Mathematics · Textbook solutions

Probability

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 113 questions

13.2 Conditional Probability

34 q

Solved Examples

Worked · 7

13.1 Eg.1 $P(A) = \frac{7}{13}$
13.1 Eg.2 $A family has two children. What is the probability that both the children are boys given that at least one of them is a boy?$
13.1 Eg.3 $Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?$
13.1 Eg.4 $In a school, there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in Class XII given that the chosen student is a girl?$
13.1 Eg.5 $A die is thrown three times. Events A and B are defined as below: A : 4 on the third throw B : 6 on the first and 5 on the second throw Find the probability of A given that B has already occurred.$
13.1 Eg.6 $A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?$
13.1 Eg.7 $Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that 'the die shows a number greater than 4' given that 'there is at least one tail'.$

Exercise 13.1

Practice · 27

Ex 13.1 Q1 $P(E) = 0.6$
Ex 13.1 Q2 $P(A|B)$
If $P(A) = 0.8$ , $P(B) = 0.5$ and $P(B|A) = 0.4$ , find
Ex 13.1 Q3(i) $P(A \cap B)$
Ex 13.1 Q3(ii) $P(A|B)$
Ex 13.1 Q3(iii) $P(A \cup B)$
Ex 13.1 Q4 $P(A \cup B)$
If $P(A) = \frac{6}{11}$ , $P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$ , find
Ex 13.1 Q5(i) $P(A \cap B)$
Ex 13.1 Q5(ii) $P(A|B)$
Ex 13.1 Q5(iii) $P(B|A)$
Determine $P(E|F)$ . A coin is tossed three times, where
Ex 13.1 Q6(i) $E : head on third toss, F : heads on first two tosses$
Ex 13.1 Q6(ii) $E : at least two heads, F : at most two heads$
Ex 13.1 Q6(iii) $E : at most two tails, F : at least one tail$
Determine $P(E|F)$ . Two coins are tossed once, where
Ex 13.1 Q7(i) $E : tail appears on one coin, F : one coin shows head$
Ex 13.1 Q7(ii) $E : no tail appears, F : no head appears$
Ex 13.1 Q8 $P(E|F)$
Ex 13.1 Q9 $P(E|F)$
A black and a red dice are rolled.
Ex 13.1 Q10(a) $Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.$
Ex 13.1 Q10(b) $Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.$
A fair die is rolled. Consider events $E = \{1,3,5\}$ , $F = \{2,3\}$ and $G = \{2,3,4,5\}$ . Find
Ex 13.1 Q11(i) $P(E|F)$
Ex 13.1 Q11(ii) $P(E|G)$
Ex 13.1 Q11(iii) $P((E \cup F)|G)$
Ex 13.1 Q12 $Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?$
Ex 13.1 Q13 $An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?$
Ex 13.1 Q14 $Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.$
Ex 13.1 Q15 $Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event 'the coin shows a tail', given that 'at least one die shows a 3'.$
Ex 13.1 Q16 $P(A) = \frac{1}{2}$
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Ex 13.1 Q17 $P(A|B) = P(B|A)$
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13.3 Multiplication Theorem on Probability and 13.4 Independent Events

39 q

Solved Examples

Worked · 7

13.2 Eg.8 $An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?$
13.2 Eg.9 $Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace?$
13.2 Eg.10 $A die is thrown. If E is the event 'the number appearing is a multiple of 3' and F be the event 'the number appearing is even' then find whether E and F are independent ?$
13.2 Eg.11 $An unbiased die is thrown twice. Let the event A be 'odd number on the first throw' and B the event 'odd number on the second throw'. Check the independence of the events A and B.$
13.2 Eg.12 $Three coins are tossed simultaneously. Consider the event E 'three heads or three tails', F 'at least two heads' and G 'at most two heads'. Of the pairs (E,F), (E,G) and (F,G), which are independent? which are dependent?$
13.2 Eg.13 $F'$
13.2 Eg.14 $1 - P(A')\,P(B')$

Exercise 13.2

Practice · 32

Ex 13.2 Q1 $P(A) = \frac{3}{5}$
Ex 13.2 Q2 $Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.$
Ex 13.2 Q3 $A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.$
Ex 13.2 Q4 $A fair coin and an unbiased die are tossed. Let A be the event 'head appears on the coin' and B be the event '3 on the die'. Check whether A and B are independent events or not.$
Ex 13.2 Q5 $A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, 'the number is even,' and B be the event, 'the number is red'. Are A and B independent?$
Ex 13.2 Q6 $P(E) = \frac{3}{5}$
Given that the events A and B are such that $P(A) = \frac{1}{2}$ , $P(A \cup B) = \frac{3}{5}$ and $P(B) = p$ . Find $p$ if they are
Ex 13.2 Q7(i) $mutually exclusive$
Ex 13.2 Q7(ii) $independent$
Let A and B be independent events with $P(A) = 0.3$ and $P(B) = 0.4$ . Find
Ex 13.2 Q8(i) $P(A \cap B)$
Ex 13.2 Q8(ii) $P(A \cup B)$
Ex 13.2 Q8(iii) $P(A|B)$
Ex 13.2 Q8(iv) $P(B|A)$
Ex 13.2 Q9 $P(A) = \frac{1}{4}$
Ex 13.2 Q10 $P(A) = \frac{1}{2}$
Given two independent events A and B such that $P(A) = 0.3$ , $P(B) = 0.6$ . Find
Ex 13.2 Q11(i) $P(\text{A and B})$
Ex 13.2 Q11(ii) $P(\text{A and not B})$
Ex 13.2 Q11(iii) $P(\text{A or B})$
Ex 13.2 Q11(iv) $P(\text{neither A nor B})$
Ex 13.2 Q12 $A die is tossed thrice. Find the probability of getting an odd number at least once.$
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
Ex 13.2 Q13(i) $both balls are red.$
Ex 13.2 Q13(ii) $first ball is black and second is red.$
Ex 13.2 Q13(iii) $one of them is black and other is red.$
Probability of solving specific problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that
Ex 13.2 Q14(i) $the problem is solved$
Ex 13.2 Q14(ii) $exactly one of them solves the problem.$
One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent ?
Ex 13.2 Q15(i) $E : 'the card drawn is a spade' F : 'the card drawn is an ace'$
Ex 13.2 Q15(ii) $E : 'the card drawn is black' F : 'the card drawn is a king'$
Ex 13.2 Q15(iii) $E : 'the card drawn is a king or queen' F : 'the card drawn is a queen or jack'.$
In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random.
Ex 13.2 Q16(a) $Find the probability that she reads neither Hindi nor English newspapers.$
Ex 13.2 Q16(b) $If she reads Hindi newspaper, find the probability that she reads English newspaper.$
Ex 13.2 Q16(c) $If she reads English newspaper, find the probability that she reads Hindi newspaper.$
Ex 13.2 Q17 $The probability of obtaining an even prime number on each die, when a pair of dice is rolled is$
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Ex 13.2 Q18 $Two events A and B will be independent, if$
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13.5 Bayes' Theorem

21 q

Solved Examples

Worked · 7

13.3 Eg.15 $A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike, and 0.32 that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time.$
13.3 Eg.16 $Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II.$
13.3 Eg.17 $Given three identical boxes I, II and III, each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in the box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?$
13.3 Eg.18 $-$
13.3 Eg.19 $In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?$
13.3 Eg.20 $\frac{3}{10}, \frac{1}{5}, \frac{1}{10}$
13.3 Eg.21 $A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.$

Exercise 13.3

Practice · 14

Ex 13.3 Q1 $An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?$
Ex 13.3 Q2 $A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.$
Ex 13.3 Q3 $Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?$
Ex 13.3 Q4 $\frac{3}{4}$
Ex 13.3 Q5 $A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?$
Ex 13.3 Q6 $There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?$
Ex 13.3 Q7 $An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?$
Ex 13.3 Q8 $A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?$
Ex 13.3 Q9 $Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.$
Ex 13.3 Q10 $Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?$
Ex 13.3 Q11 $A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?$
Ex 13.3 Q12 $A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.$
Ex 13.3 Q13 $\frac{4}{5}$
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Ex 13.3 Q14 $A \subset B$
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Miscellaneous Exercise on Chapter 13

19 q

Solved Examples

Worked · 3

Misc Eg.22 $Coloured balls are distributed in four boxes as shown in the following table: Box Black White Red Blue I 3456 II 2222 III 1231 IV 4315 A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III?$
Misc Eg.23 $A and B throw a die alternatively till one of them gets a '6' and wins the game. Find their respective probabilities of winning, if A starts first.$
Misc Eg.24 $If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up, it produces only 40% acceptable items. Past experience shows that 80% of the set ups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly setup.$

Box	Black	White	Red	Blue
I	3	4	5	6
II	2	2	2	2
III	1	2	3	1
IV	4	3	1	5

Miscellaneous Exercise

Practice · 16

A and B are two events such that $P(A) \neq 0$ . Find $P(B|A)$ , if
Misc Q1(i) $A is a subset of B$
Misc Q1(ii) $A \cap B = \phi$
A couple has two children,
Misc Q2(i) $Find the probability that both children are males, if it is known that at least one of the children is male.$
Misc Q2(ii) $Find the probability that both children are females, if it is known that the elder child is a female.$
Misc Q3 $Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.$
Misc Q4 $Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?$
Misc Q5 $If a leap year is selected at random, what is the chance that it will contain 53 tuesdays?$
Misc Q6 $Suppose we have four boxes A,B,C and D containing coloured marbles as given below: Box Red White Black A 163 B 622 C 811 D 064 One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C?$
Misc Q7 $Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?$
Misc Q8 $\frac{1}{2}$
An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known: P(A fails) = 0.2 P(B fails alone) = 0.15 P(A and B fail) = 0.15 Evaluate the following probabilities
Misc Q9(i) $P(A fails|B has failed)$
Misc Q9(ii) $P(A fails alone)$
Misc Q10 $Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.$
Misc Q11 $P(A) \neq 0$
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Misc Q12 $P(A|B) > P(A)$
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Misc Q13 $P(A) + P(B) - P(A \text{ and } B) = P(A)$
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Box	Red	White	Black
A	1	6	3
B	6	2	2
C	8	1	1
D	0	6	4

What the board has asked from this chapter

Questions per paper · 26 papers · 2022 to 2026

Bars average the full papers of each year. 33 further questions come from set variants and are listed below.

2026

33 q

18 $\dfrac{4}{5}$
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Independent Events
18 $3P(A) = P(B) = \dfrac{3}{5}$
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Conditional Probability
18 $The probability that it will rain tomorrow in cities A, B and C is 60%, 70% and 80% respectively. The probability that it will rain tomorrow in at least one of the cities is :$
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Independent Events
18 $P(E) = \dfrac{3}{10}$
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Independent Events
18 $A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red ?$
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Independent Events
18 $P(A) \neq 0$
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Conditional Probability
Direction : Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
19 $\frac{2}{3}$
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Conditional Probability
31 (a) $The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.$
Independent Events
31 (a) $A = \{1,\ 2,\ 5\}$
Conditional Probability
31 (a) $Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.$
Theorem of Total Probability
31 (b) $P(X') + P(Y') = 2 - 2a + b$
Conditional Probability
31 (b) $A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected. Find P(A and B) and find whether the events A and B are independent events or not.$
Independent Events
31 $A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that, Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that : (i) exactly one surgery is successful, (ii) at most two surgeries are successful.$
Independent Events
31 $\dfrac{1}{3}$
Theorem of Total Probability
31 (b) $P(E/F)$
Conditional Probability
Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information, answer the following questions :
36 (i) $What is the probability that selected person is a female ?$
Conditional Probability
Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information, answer the following questions :
36 (ii) $If a male person is selected, what is the probability that he will not be suffering from lung problems ?$
Conditional Probability
Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information, answer the following questions :
36 (iii) (a) $A person selected at random is detected with lung complications. Find the probability that selected person is a female.$
Bayes' Theorem
Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information, answer the following questions :
36 (iii) (b) $A person selected at random is not having lung problems, find the probability that the person is a male.$
Bayes' Theorem
There are three types of vaccines $A_1$ , $A_2$ , $A_3$ , available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine $A_1$ , 35% of the population was given Vaccine $A_2$ and 40% of the population was given Vaccine $A_3$ . The survey also stated that the probabilities that Vaccines $A_1$ , $A_2$ and $A_3$ would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, answer the following questions : Find the probability that :
37 (i) $A_2$
Conditional Probability
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII. Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination. Based on the above information, answer the following questions.
37 (i) $Find the probability that a student selected at random is a regular student.$
Conditional Probability
In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third prizes of ₹ 50,000 each. A total of 1,00,000 jackpot tickets each costing ₹ 100 were sold there by raising a fund of ₹ 1,00,00,000. Rohan bought one ticket. Based on given information, answer the following questions :
37 (i) $What are the possible amounts, the person can win ?$
Conditional Probability
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII. Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination. Based on the above information, answer the following questions.
37 (ii) $A student is selected at random from a group of dropout students. What is the probability that the student will not qualify the examination ?$
Conditional Probability
In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third prizes of ₹ 50,000 each. A total of 1,00,000 jackpot tickets each costing ₹ 100 were sold there by raising a fund of ₹ 1,00,00,000. Rohan bought one ticket. Based on given information, answer the following questions :
37 (ii) (a) $What is the probability that the person wins atleast ₹ 2,00,000 ?$
Conditional Probability
There are three types of vaccines $A_1$ , $A_2$ , $A_3$ , available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine $A_1$ , 35% of the population was given Vaccine $A_2$ and 40% of the population was given Vaccine $A_3$ . The survey also stated that the probabilities that Vaccines $A_1$ , $A_2$ and $A_3$ would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, answer the following questions : Find the probability that :
37 (ii) $If a person is chosen randomly, he/she will be protected from the infection.$
Theorem of Total Probability
There are three types of vaccines $A_1$ , $A_2$ , $A_3$ , available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine $A_1$ , 35% of the population was given Vaccine $A_2$ and 40% of the population was given Vaccine $A_3$ . The survey also stated that the probabilities that Vaccines $A_1$ , $A_2$ and $A_3$ would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, answer the following questions : Find the probability that :
37 (iii) (a) $A_1$
Bayes' Theorem
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII. Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination. Based on the above information, answer the following questions.
37 (iii) (a) $A student selected at random qualified the examination. Find the probability that student is not a dropout.$
Bayes' Theorem
In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third prizes of ₹ 50,000 each. A total of 1,00,000 jackpot tickets each costing ₹ 100 were sold there by raising a fund of ₹ 1,00,00,000. Rohan bought one ticket. Based on given information, answer the following questions :
37 (ii) (b) $What is the probability that the person does not win any amount ?$
Conditional Probability
A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII. Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination. Based on the above information, answer the following questions.
37 (iii) (b) $A student selected at random did not qualify the examination. Find the probability that the student was a regular student.$
Bayes' Theorem
There are three types of vaccines $A_1$ , $A_2$ , $A_3$ , available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine $A_1$ , 35% of the population was given Vaccine $A_2$ and 40% of the population was given Vaccine $A_3$ . The survey also stated that the probabilities that Vaccines $A_1$ , $A_2$ and $A_3$ would protect against the infection were 60%, 55% and 50% respectively. Based on the above information, answer the following questions : Find the probability that :
37 (iii) (b) $A_3$
Bayes' Theorem
In an online jackpot, there is one first prize of ₹ 3,00,000, two second prizes of ₹ 2,00,000 each and three third prizes of ₹ 50,000 each. A total of 1,00,000 jackpot tickets each costing ₹ 100 were sold there by raising a fund of ₹ 1,00,00,000. Rohan bought one ticket. Based on given information, answer the following questions :
37 (iii) $In another jackpot, Rohan also bought a ticket having a prize money of ₹ 5,00,000. The chances of winning the jackpot are 1 in 1,00,000. Find the probability that on exactly one of tickets he wins the jackpot.$
Independent Events
An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is $\dfrac{i}{6}$ , where i = 1, 2, 3. Based on the above information, answer the following questions : A person selects a cap.
38 (i) $What is the probability that he selects a red cap ?$
Theorem of Total Probability
An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is $\dfrac{i}{6}$ , where i = 1, 2, 3. Based on the above information, answer the following questions : A person selects a cap.
38 (ii) $If he selects a green cap, what is the probability that the cap has come from Box II ?$
Bayes' Theorem

2025

47 q

2 $P(A \cup B) = 0{\cdot}9$
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Conditional Probability
7 $P(E) = \frac{2}{3}$
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Independent Events
7 $P(M \cap N) = 0.5$
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Conditional Probability
7 $P(E) > 0$
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Conditional Probability
11 $-4$
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Random Variables and Probability Distributions
11 $In the following probability distribution, the value of p is : X 0123 P(X) p p 0.3 2p$
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Random Variables and Probability Distributions
17 $P(B) = \dfrac{1}{5}$
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Conditional Probability
17 $P(A) = \dfrac{1}{7}$
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Conditional Probability
17 $P(A) = \dfrac{1}{5}$
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Conditional Probability
18 $A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is :$
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Independent Events
18 $Chances that three persons A, B, and C go to the market are 30%, 60% and 50% respectively. The probability that at least one will go to the market is :$
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Independent Events
18 $0{\cdot}10$
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Independent Events
18 $A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is :$
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Independent Events
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
19 $P(A \cap B) = 0$
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Independent Events
25 (a) $10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.$
Random Variables and Probability Distributions
25 (b) $In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village ?$
Conditional Probability
29 (a) $\frac{p}{3}$
Random Variables and Probability Distributions
29 (b) $In a city, a survey was conducted among residents about their preferred mode of commuting. It was found that 50% people preferred using public transport, 35% preferred using a bicycle and 20% use both public transport and a bicycle. If a person is selected at random, find the probability that : (i) The person uses only public transport. (ii) The person uses a bicycle, given that they also use the public transport. (iii) The person uses neither public transport nor a bicycle.$
Conditional Probability
29 (a) $Consider the experiment of tossing a coin. If the coin shows head, toss it again; but if it shows a tail, then throw a die. Find the conditional probability of the event A : 'the die shows a number greater than 3' given that B : 'there is at least one tail'.$
Conditional Probability
29 (a) $Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.$
Random Variables and Probability Distributions
30 (a) $The probability distribution for the number of students being absent in a class on a Saturday is as follows : X 0245 P(X) p 2p 3p p Where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday.$
Random Variables and Probability Distributions
29 (b) $A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.$
Random Variables and Probability Distributions
29 (b) $2\lambda$
Random Variables and Probability Distributions
30 (a) $P(2) = \dfrac{3}{10}$
Random Variables and Probability Distributions
30 (b) $A = \{(x, y) : x + y = 9\}$
Independent Events
30 (b) $For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.$
Theorem of Total Probability
31 (a) $0{\cdot}7$
Conditional Probability
31 $0{\cdot}03$
Independent Events
31 $Bag I contains 4 white and 5 black balls. Bag II contains 6 white and 7 black balls. A ball drawn randomly by from bag I is transferred to bag II and then a ball is drawn randomly from bag II. Find the probability that the ball drawn is white.$
Theorem of Total Probability
31 (b) $A person has a fruit box that contains 6 apples and 4 oranges. He picks out a fruit three times, one after the other, after replacing the previous one in the box. Find : (i) The probability distribution of the number of oranges he draws. (ii) The expectation of the random variable (number of oranges).$
Random Variables and Probability Distributions
31 $A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed three times, find the probability distribution of number of tails. Hence, find the mean of the distribution.$
Random Variables and Probability Distributions
Some students are having a misconception while comparing decimals. For example, a student may mention that $78{\cdot}56 > 78{\cdot}9$ as 7856 > 789. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question : In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table :
Name of student Distance of javelin (in meters)
Ajay $47{\cdot}7$
Bijoy $47{\cdot}07$
Kartik $43{\cdot}09$
Dinesh $43{\cdot}9$
Devesh $45{\cdot}2$
The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. 80% of the students having misconception answered Bijoy as the correct answer in the paper. 90% of the students who are identified with not having misconception, did not answer Bijoy as their answer. On the basis of the above information, answer the following questions :
36 (i) $What is the probability of a student not having misconception but still answers Bijoy in the test ?$
Multiplication Theorem on Probability
Some students are having a misconception while comparing decimals. For example, a student may mention that $78{\cdot}56 > 78{\cdot}9$ as 7856 > 789. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question : In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table :
Name of student Distance of javelin (in meters)
Ajay $47{\cdot}7$
Bijoy $47{\cdot}07$
Kartik $43{\cdot}09$
Dinesh $43{\cdot}9$
Devesh $45{\cdot}2$
The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. 80% of the students having misconception answered Bijoy as the correct answer in the paper. 90% of the students who are identified with not having misconception, did not answer Bijoy as their answer. On the basis of the above information, answer the following questions :
36 (ii) $What is the probability that a randomly selected student answers Bijoy as his answer in the test ?$
Theorem of Total Probability
Some students are having a misconception while comparing decimals. For example, a student may mention that $78{\cdot}56 > 78{\cdot}9$ as 7856 > 789. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question : In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table :
Name of student Distance of javelin (in meters)
Ajay $47{\cdot}7$
Bijoy $47{\cdot}07$
Kartik $43{\cdot}09$
Dinesh $43{\cdot}9$
Devesh $45{\cdot}2$
The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. 80% of the students having misconception answered Bijoy as the correct answer in the paper. 90% of the students who are identified with not having misconception, did not answer Bijoy as their answer. On the basis of the above information, answer the following questions :
36 (iii) (a) $What is the probability that a student who answered as Bijoy is having misconception ?$
Bayes' Theorem
Some students are having a misconception while comparing decimals. For example, a student may mention that $78{\cdot}56 > 78{\cdot}9$ as 7856 > 789. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question : In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table :
Name of student Distance of javelin (in meters)
Ajay $47{\cdot}7$
Bijoy $47{\cdot}07$
Kartik $43{\cdot}09$
Dinesh $43{\cdot}9$
Devesh $45{\cdot}2$
The students were asked to identify who has thrown the javelin the farthest. Based on the test attempted by the students, the teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. 80% of the students having misconception answered Bijoy as the correct answer in the paper. 90% of the students who are identified with not having misconception, did not answer Bijoy as their answer. On the basis of the above information, answer the following questions :
36 (iii) (b) $What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception ?$
Bayes' Theorem
Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated). Based on the above, answer the following :
37 (i) (a) $What is the probability that a randomly selected car is an electric car ?$
Theorem of Total Probability
Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated). Based on the above, answer the following :
37 (i) (b) $What is the probability that a randomly selected car is a petrol car ?$
Theorem of Total Probability
Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated). Based on the above, answer the following :
37 (ii) $A car is selected at random and is found to be electric. What is the probability that it was manufactured by Comet ?$
Bayes' Theorem
Three persons viz. Amber, Bonzi and Comet are manufacturing cars which run on petrol and on battery as well. Their production share in the market is 60%, 30% and 10% respectively. Of their respective production capacities, 20%, 10% and 5% cars respectively are electric (or battery operated). Based on the above, answer the following :
37 (iii) $A car is selected at random and is found to be electric. What is the probability that it was manufactured by Amber or Bonzi ?$
Bayes' Theorem
A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C. A person buys a smartphone from this shop.
38 (i) $Find the probability that it was defective.$
Theorem of Total Probability
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record. Let $A_1$ : People with good health, $A_2$ : People with average health, and $A_3$ : People with poor health. During a pandemic, the data expressed that the chances of people contracting the disease from category $A_1$ , $A_2$ and $A_3$ are 25%, 35% and 50%, respectively. Based upon the above information, answer the following questions :
38 (i) $A person was tested randomly. What is the probability that he/she has contracted the disease ?$
Theorem of Total Probability
A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage plant and 8 seeds of radish plant. The shopkeeper assured him of germination probabilities of brinjal, cabbage and radish to be 25%, 35% and 40% respectively. But before he could plant the seeds, they got mixed up in the bag and he had to sow them randomly. Based upon the above information, answer the following questions :
38 (i) $Calculate the probability of a randomly chosen seed to germinate.$
Theorem of Total Probability
A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively. Based on the above information, answer the following :
38 (i) $What is the probability that a customer after availing the loan will default on the loan repayment ?$
Theorem of Total Probability
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record. Let $A_1$ : People with good health, $A_2$ : People with average health, and $A_3$ : People with poor health. During a pandemic, the data expressed that the chances of people contracting the disease from category $A_1$ , $A_2$ and $A_3$ are 25%, 35% and 50%, respectively. Based upon the above information, answer the following questions :
38 (ii) $A_2$
Bayes' Theorem
A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C. A person buys a smartphone from this shop.
38 (ii) $What is the probability that this defective smartphone was manufactured by company B ?$
Bayes' Theorem
A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively. Based on the above information, answer the following :
38 (ii) $A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest ?$
Bayes' Theorem
A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage plant and 8 seeds of radish plant. The shopkeeper assured him of germination probabilities of brinjal, cabbage and radish to be 25%, 35% and 40% respectively. But before he could plant the seeds, they got mixed up in the bag and he had to sow them randomly. Based upon the above information, answer the following questions :
38 (ii) $What is the probability that it is a cabbage seed, given that the chosen seed germinates ?$
Bayes' Theorem

X	$-4$	$-3$	$-2$	$-1$	$0$
P(X)	$0.1$	$0.2$	$0.3$	$0.2$	$0.2$

X	0	1	2	3
P(X)	p	p	0.3	2p

X	0	1	2	3
P(X)	p	$\frac{p}{3}$	$\frac{p}{6}$	$\frac{p}{12}$

X	0	2	4	5
P(X)	p	2p	3p	p

X	1	2	3	$2\lambda$	$3\lambda$	$4\lambda$
P(X)	$\frac{11}{30}$	$\frac{1}{15}$	$\frac{1}{10}$	$\frac{3}{10}$	$\frac{1}{15}$	$\frac{1}{10}$

Name of student	Distance of javelin (in meters)
Ajay	$47{\cdot}7$
Bijoy	$47{\cdot}07$
Kartik	$43{\cdot}09$
Dinesh	$43{\cdot}9$
Devesh	$45{\cdot}2$

Name of student	Distance of javelin (in meters)
Ajay	$47{\cdot}7$
Bijoy	$47{\cdot}07$
Kartik	$43{\cdot}09$
Dinesh	$43{\cdot}9$
Devesh	$45{\cdot}2$

Name of student	Distance of javelin (in meters)
Ajay	$47{\cdot}7$
Bijoy	$47{\cdot}07$
Kartik	$43{\cdot}09$
Dinesh	$43{\cdot}9$
Devesh	$45{\cdot}2$

Name of student	Distance of javelin (in meters)
Ajay	$47{\cdot}7$
Bijoy	$47{\cdot}07$
Kartik	$43{\cdot}09$
Dinesh	$43{\cdot}9$
Devesh	$45{\cdot}2$

2024

31 q

11 $P(S \,|\, E) =$
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Conditional Probability
11 $P(E) = 0{\cdot}1$
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Conditional Probability
11 $P(A|B) = P(A'|B)$
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Conditional Probability
17 $The probability distribution of a random variable X is : X 01234 P(X) 0.1 k 2k k 0.1 where k is some unknown constant. The probability that the random variable X takes the value 2 is :$
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Random Variables and Probability Distributions
18 $P(A/B) = P(B/A) \ne 0$
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Conditional Probability
31 (a) $A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.$
Bayes' Theorem
30 $A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X.$
Random Variables and Probability Distributions
31 $P(\overline{E}) = 0{\cdot}6$
Independent Events
31 (b) $A biased die is twice as likely to show an even number as an odd number. If such a die is thrown twice, find the probability distribution of the number of sixes. Also, find the mean of the distribution.$
Random Variables and Probability Distributions
31 $Bag I contains 3 red and 4 black balls, Bag II contains 5 red and 2 black balls. Two balls are transferred at random from Bag I to Bag II and then a ball is drawn at random from Bag II. Find the probability that the drawn ball is red in colour.$
Theorem of Total Probability
31 $0{\cdot}3$
Bayes' Theorem
31 $P(X \ge 3)$
Random Variables and Probability Distributions
31 $An urn contains 3 red and 2 white marbles. Two marbles are drawn one by one with replacement from the urn. Find the probability distribution of the number of white balls. Also, find the mean of the number of white balls drawn.$
Random Variables and Probability Distributions
Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below : $P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3 \\ 2kx, & \text{for } x = 4, 5, 6 \\ 0, & \text{otherwise} \end{cases}$ where x denotes the number of hours. Based on the above information, answer the following questions :
36 (i) $Express the probability distribution given above in the form of a probability distribution table.$
Random Variables and Probability Distributions
Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below : $P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3 \\ 2kx, & \text{for } x = 4, 5, 6 \\ 0, & \text{otherwise} \end{cases}$ where x denotes the number of hours. Based on the above information, answer the following questions :
36 (ii) $Find the value of k.$
Random Variables and Probability Distributions
Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below : $P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3 \\ 2kx, & \text{for } x = 4, 5, 6 \\ 0, & \text{otherwise} \end{cases}$ where x denotes the number of hours. Based on the above information, answer the following questions :
36 (iii) (a) $Find the mean number of hours spent by the student.$
Random Variables and Probability Distributions
Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves. A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below : $P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3 \\ 2kx, & \text{for } x = 4, 5, 6 \\ 0, & \text{otherwise} \end{cases}$ where x denotes the number of hours. Based on the above information, answer the following questions :
36 (iii) (b) $P(1 < X < 6)$
Random Variables and Probability Distributions
According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55%, 37% and 17% due to severe, moderate and light turbulence respectively. On the basis of the above information, answer the following questions :
37 (i) $Find the probability that an airplane reached its destination late.$
Theorem of Total Probability
According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that, an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are 55%, 37% and 17% due to severe, moderate and light turbulence respectively. On the basis of the above information, answer the following questions :
37 (ii) $If the airplane reached its destination late, find the probability that it was due to moderate turbulence.$
Bayes' Theorem
A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. Based on the above information, answer the following questions :
38 (i) $\mathrm{E}_1$
Conditional Probability
Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is $0{\cdot}00001\%$ . Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let $E_1$ be the event that there is a plane crash and $E_2$ be the event that there is no crash. Let A be the event that passengers survive after the journey. On the basis of the above information, answer the following questions :
38 (i) $Find the probability that the airplane will not crash.$
Conditional Probability
A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. Based on the above information, answer the following questions :
38 (ii) $P(\mathrm{A}|\mathrm{E}_1)$
Conditional Probability
Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is $\dfrac{1}{5}$ , Jaspreet's selection is $\dfrac{1}{3}$ and Alia's selection is $\dfrac{1}{4}$ . The event of selection is independent of each other. Based on the above information, answer the following questions :
38 (i) $What is the probability that at least one of them is selected ?$
Independent Events
Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is $0{\cdot}00001\%$ . Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let $E_1$ be the event that there is a plane crash and $E_2$ be the event that there is no crash. Let A be the event that passengers survive after the journey. On the basis of the above information, answer the following questions :
38 (ii) $P(A \,|\, E_1) + P(A \,|\, E_2)$
Conditional Probability
Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is $\dfrac{1}{5}$ , Jaspreet's selection is $\dfrac{1}{3}$ and Alia's selection is $\dfrac{1}{4}$ . The event of selection is independent of each other. Based on the above information, answer the following questions :
38 (ii) $P(G\,|\,\overline{H})$
Conditional Probability
Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is $0{\cdot}00001\%$ . Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let $E_1$ be the event that there is a plane crash and $E_2$ be the event that there is no crash. Let A be the event that passengers survive after the journey. On the basis of the above information, answer the following questions :
38 (iii) (a) $Find P(A).$
Theorem of Total Probability
A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. Based on the above information, answer the following questions :
38 (iii) (a) $Find the probability of customer paying second month's bill in time.$
Theorem of Total Probability
A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. Based on the above information, answer the following questions :
38 (iii) (b) $Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.$
Bayes' Theorem
Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is $\dfrac{1}{5}$ , Jaspreet's selection is $\dfrac{1}{3}$ and Alia's selection is $\dfrac{1}{4}$ . The event of selection is independent of each other. Based on the above information, answer the following questions :
38 (iii) (a) $Find the probability that exactly one of them is selected.$
Independent Events
Airplanes are by far the safest mode of transportation when the number of transported passengers are measured against personal injuries and fatality totals. Previous records state that the probability of an airplane crash is $0{\cdot}00001\%$ . Further, there are 95% chances that there will be survivors after a plane crash. Assume that in case of no crash, all travellers survive. Let $E_1$ be the event that there is a plane crash and $E_2$ be the event that there is no crash. Let A be the event that passengers survive after the journey. On the basis of the above information, answer the following questions :
38 (iii) (b) $P(E_2 \,|\, A)$
Bayes' Theorem
Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is $\dfrac{1}{5}$ , Jaspreet's selection is $\dfrac{1}{3}$ and Alia's selection is $\dfrac{1}{4}$ . The event of selection is independent of each other. Based on the above information, answer the following questions :
38 (iii) (b) $Find the probability that exactly two of them are selected.$
Independent Events

X	0	1	2	3	4
P(X)	0.1	k	2k	k	0.1

X	1	2	3	4	5
P(X)	0.2	a	a	0.2	b

2023

43 q

3 $P(A - B) = \frac{1}{5}$
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Conditional Probability
3 $Ashima can hit a target 2 out of 3 times. She tried to hit the target twice. The probability that she missed the target exactly once is$
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Independent Events
5 $P(X \cap \overline{Y}) = \frac{2}{5}$
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Independent Events
8 $P(A/B) = 2 \times P(B/A)$
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Conditional Probability
10 $P(\overline{A}) = \frac{1}{2}$
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Conditional Probability
12 $If the sum of numbers obtained on throwing a pair of dice is 9, then the probability that number obtained on one of the dice is 4, is :$
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Conditional Probability
12 $P\left(\frac{A}{B}\right) = 0{\cdot}3$
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Conditional Probability
14 $P(A \cap B) = \frac{1}{8}$
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Conditional Probability
16 $P(A) = 0{\cdot}4$
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Multiplication Theorem on Probability
17 $P(E) = 0{\cdot}3$
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Independent Events
17 $P(A) = \dfrac{4}{5}$
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Conditional Probability
18 $P(A) = \dfrac{1}{3}$
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Independent Events
18 $\dfrac{4}{5}$
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Independent Events
18 $Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is$
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Independent Events
In the following questions 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices :
19 $\dfrac{1}{3}$
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Conditional Probability
27 $Two fair dice are thrown simultaneously. If X denotes the number of sixes, find the mean of X.$
Random Variables and Probability Distributions
30 (a) $\frac{k}{2}$
Random Variables and Probability Distributions
30 (a) $Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable.$
Random Variables and Probability Distributions
30 (b) $P(A \cap \overline{B}) = \frac{1}{4}$
Independent Events
31 (a) $Two numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of two numbers selected, find the probability distribution of X.$
Random Variables and Probability Distributions
30 (b) $A and B throw a die alternately till one of them gets a '6' and wins the game. Find their respective probabilities of wining, if A starts the game first.$
Independent Events
31 (a) $A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of X.$
Random Variables and Probability Distributions
31 (b) $A fair coin and an unbiased die are tossed. Let A be the event, "Head appears on the coin" and B be the event, "3 comes on the die". Find whether A and B are independent events or not.$
Independent Events
31 (b) $There are two coins. One of them is a biased coin such that P (head) : P (tail) is 1 : 3 and the other coin is a fair coin. A coin is selected at random and tossed once. If the coin showed head, then find the probability that it is a biased coin.$
Bayes' Theorem
31 $From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with replacement. Find the probability distribution of the number of defective bulbs and hence find the mean number of defective bulbs.$
Random Variables and Probability Distributions
35 (a) $\dfrac{3}{5}$
Bayes' Theorem
35 (b) $A box contains 10 tickets, 2 of which carry a prize of ₹ 8 each, 5 of which carry a prize of ₹ 4 each, and remaining 3 carry a prize of ₹ 2 each. If one ticket is drawn at random, find the mean value of the prize.$
Random Variables and Probability Distributions
There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below : The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is $0{\cdot}44$ . On the basis of the above information, answer the following questions :
36 (i) $Find the value of x.$
Conditional Probability
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X.
X : 1 2 3 4 5 6 7 8
P(X) : p 2p 2p p 2p $p^2$ $2p^2$ $7p^2 + p$
36 (i) $Find the value of p.$
Random Variables and Probability Distributions
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X.
X : 1 2 3 4 5 6 7 8
P(X) : p 2p 2p p 2p $p^2$ $2p^2$ $7p^2 + p$
36 (ii) $Find P(X > 6).$
Random Variables and Probability Distributions
There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below : The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is $0{\cdot}44$ . On the basis of the above information, answer the following questions :
36 (ii) $Find the value of y.$
Conditional Probability
There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below : The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is $0{\cdot}44$ . On the basis of the above information, answer the following questions :
36 (iii) (a) $P\left(\frac{C}{B}\right)$
Conditional Probability
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X.
X : 1 2 3 4 5 6 7 8
P(X) : p 2p 2p p 2p $p^2$ $2p^2$ $7p^2 + p$
36 (iii) (a) $Find P(X = 3m), where m is a natural number.$
Random Variables and Probability Distributions
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X.
X : 1 2 3 4 5 6 7 8
P(X) : p 2p 2p p 2p $p^2$ $2p^2$ $7p^2 + p$
36 (iii) (b) $Find the mean E(X).$
Random Variables and Probability Distributions
There are different types of Yoga which involve the usage of different poses of Yoga Asanas, Meditation and Pranayam as shown in the figure below : The Venn diagram below represents the probabilities of three different types of Yoga, A, B and C performed by the people of a society. Further, it is given that probability of a member performing type C Yoga is $0{\cdot}44$ . On the basis of the above information, answer the following questions :
36 (iii) (b) $Find the probability that a randomly selected person of the society does Yoga of type A or B but not C.$
Conditional Probability
A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is $0{\cdot}65$ . The probability that many are not present and still the work gets completed on time is $0{\cdot}35$ . The probability that work will be completed on time when all workers are present is $0{\cdot}80$ . Let : $E_1$ : represent the event when many workers were not present for the job; $E_2$ : represent the event when all workers were present; and E : represent completing the construction work on time. Based on the above information, answer the following questions :
37 (i) $What is the probability that all the workers are present for the job ?$
Conditional Probability
Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows : A : When both father and mother are left handed : Chances of left handed child is 24%. B : When father is right handed and mother is left handed : Chances of left handed child is 22%. C : When father is left handed and mother is right handed : Chances of left handed child is 17%. D : When both father and mother are right handed : Chances of left handed child is 9%. Assuming that $P(A) = P(B) = P(C) = P(D) = \frac{1}{4}$ and L denotes the event that child is left handed. Based on the above information, answer the following questions :
37 (i) $P(L/C)$
Conditional Probability
Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows : A : When both father and mother are left handed : Chances of left handed child is 24%. B : When father is right handed and mother is left handed : Chances of left handed child is 22%. C : When father is left handed and mother is right handed : Chances of left handed child is 17%. D : When both father and mother are right handed : Chances of left handed child is 9%. Assuming that $P(A) = P(B) = P(C) = P(D) = \frac{1}{4}$ and L denotes the event that child is left handed. Based on the above information, answer the following questions :
37 (ii) $P(\overline{L}/A)$
Conditional Probability
A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is $0{\cdot}65$ . The probability that many are not present and still the work gets completed on time is $0{\cdot}35$ . The probability that work will be completed on time when all workers are present is $0{\cdot}80$ . Let : $E_1$ : represent the event when many workers were not present for the job; $E_2$ : represent the event when all workers were present; and E : represent completing the construction work on time. Based on the above information, answer the following questions :
37 (ii) $What is the probability that construction will be completed on time ?$
Theorem of Total Probability
Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows : A : When both father and mother are left handed : Chances of left handed child is 24%. B : When father is right handed and mother is left handed : Chances of left handed child is 22%. C : When father is left handed and mother is right handed : Chances of left handed child is 17%. D : When both father and mother are right handed : Chances of left handed child is 9%. Assuming that $P(A) = P(B) = P(C) = P(D) = \frac{1}{4}$ and L denotes the event that child is left handed. Based on the above information, answer the following questions :
37 (iii)(a) $P(A/L)$
Bayes' Theorem
A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is $0{\cdot}65$ . The probability that many are not present and still the work gets completed on time is $0{\cdot}35$ . The probability that work will be completed on time when all workers are present is $0{\cdot}80$ . Let : $E_1$ : represent the event when many workers were not present for the job; $E_2$ : represent the event when all workers were present; and E : represent completing the construction work on time. Based on the above information, answer the following questions :
37 (iii) (a) $What is the probability that many workers are not present given that the construction work is completed on time ?$
Bayes' Theorem
A building contractor undertakes a job to construct 4 flats on a plot along with parking area. Due to strike the probability of many construction workers not being present for the job is $0{\cdot}65$ . The probability that many are not present and still the work gets completed on time is $0{\cdot}35$ . The probability that work will be completed on time when all workers are present is $0{\cdot}80$ . Let : $E_1$ : represent the event when many workers were not present for the job; $E_2$ : represent the event when all workers were present; and E : represent completing the construction work on time. Based on the above information, answer the following questions :
37 (iii) (b) $What is the probability that all workers were present given that the construction job was completed on time ?$
Bayes' Theorem
Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows : A : When both father and mother are left handed : Chances of left handed child is 24%. B : When father is right handed and mother is left handed : Chances of left handed child is 22%. C : When father is left handed and mother is right handed : Chances of left handed child is 17%. D : When both father and mother are right handed : Chances of left handed child is 9%. Assuming that $P(A) = P(B) = P(C) = P(D) = \frac{1}{4}$ and L denotes the event that child is left handed. Based on the above information, answer the following questions :
37 (iii)(b) $Find the probability that a randomly selected child is left handed given that exactly one of the parents is left handed.$
Theorem of Total Probability

X	1	2	3
P(X)	$\frac{k}{2}$	$\frac{k}{3}$	$\frac{k}{6}$

X :	1	2	3	4	5	6	7	8
P(X) :	p	2p	2p	p	2p	$p^2$	$2p^2$	$7p^2 + p$

X :	1	2	3	4	5	6	7	8
P(X) :	p	2p	2p	p	2p	$p^2$	$2p^2$	$7p^2 + p$

X :	1	2	3	4	5	6	7	8
P(X) :	p	2p	2p	p	2p	$p^2$	$2p^2$	$7p^2 + p$

X :	1	2	3	4	5	6	7	8
P(X) :	p	2p	2p	p	2p	$p^2$	$2p^2$	$7p^2 + p$

2022

25 q

1 $A bag contains 3 red and 4 white balls. Three balls are drawn at random, one-by-one without replacement from the bag. If the first ball drawn is red in colour, then find the probability that the remaining two balls drawn are also red in colour.$
Conditional Probability
2 $A coin is tossed twice. The following table shows the probability distribution of number of tails : X 012 P(X) K 6K 9K (a) Find the value of K. (b) Is the coin tossed biased or unbiased ? Justify your answer.$
Random Variables and Probability Distributions
3 $x_1$
Random Variables and Probability Distributions
3 $Three friends A, B and C got their photograph clicked. Find the probability that B is standing at the central position, given that A is standing at the left corner.$
Conditional Probability
5 $Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spade cards.$
Random Variables and Probability Distributions
4 $A pair of dice is thrown. It is given that the sum of numbers appearing on both dice is an even number. Find the probability that the number appearing on at least one die is 3.$
Conditional Probability
4 $P(A) = \frac{5}{8}$
Conditional Probability
6 (a) $P(A) = \dfrac{1}{2}$
Independent Events
5 $\frac{2}{3}$
Independent Events
6 (a) $A pair of dice is thrown and the sum of the numbers appearing on the dice is observed to be 7. Find the probability that the number 5 has appeared on atleast one die.$
Conditional Probability
5 $Two balls are drawn at random from a bag containing 2 red balls and 3 blue balls, without replacement. Let the variable X denotes the number of red balls. Find the probability distribution of X.$
Random Variables and Probability Distributions
6 (b) $\dfrac{1}{3}$
Independent Events
6 (b) $B_1$
Multiplication Theorem on Probability
6 $There are two bags. Bag I contains 1 red and 3 white balls, and Bag II contains 3 red and 5 white balls. A bag is selected at random and a ball is drawn from it. Find the probability that the ball so drawn is red in colour.$
Theorem of Total Probability
11 $Three persons A, B and C apply for a job of manager in a private company. Chances of their selection are in the ratio 1 : 2 : 4. The probability that A, B and C can introduce changes to increase the profits of a company are 0.8, 0.5 and 0.3 respectively. If increase in the profit does not take place, find the probability that it is due to the appointment of A.$
Bayes' Theorem
12 $A man is known to speak truth 7 out of 10 times. He threw a pair of dice and reports that doublet appeared. Find the probability that it was actually a doublet.$
Bayes' Theorem
12 $A card from a pack of 52 playing cards is lost. From the remaining cards, 2 cards are drawn at random without replacement, and are found to be both aces. Find the probability that lost card being an ace.$
Bayes' Theorem
13 $In a factory, machine A produces 30% of total output, machine B produces 25% and the machine C produces the remaining output. The defective items produced by machines A, B and C are 1%, 1.2%, 2% respectively. An item is picked at random from a day's output and found to be defective. Find the probability that it was produced by machine B?$
Bayes' Theorem
13 $There are two boxes, namely box-I and box-II. Box-I contains 3 red and 6 black balls. Box-II contains 5 red and 5 black balls. One of the two boxes, is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box-II.$
Bayes' Theorem
A shopkeeper sells three types of flower seeds A1, A2, A3. They are sold in the form of a mixture, where the proportions of these seeds are 4 : 4 : 2, respectively. The germination rates of the three types of seeds are 45%, 60% and 35% respectively. Based on the above information :
14 (a) $Calculate the probability that a randomly chosen seed will germinate;$
Theorem of Total Probability
In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards. Archer A is likely to earn 10 points with a probability of $0{\cdot}8$ and Archer B is likely the earn 10 points with a probability of $0{\cdot}9$ . Based on the above information, answer the following questions : If both of them hit the Archery target, then find the probability that
14 (a) $exactly one of them earns 10 points.$
Independent Events
At the start of a cricket match, a coin is tossed and the team winning the toss has the opportunity to choose to bat or bowl. Such a coin is unbiased with equal probabilities of getting head and tail. Based on the above information, answer the following questions :
14 (a) $If such a coin is tossed 2 times, then find the probability distribution of number of tails.$
Random Variables and Probability Distributions
In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are allotted points 9 to 1 in sequential order moving outwards. Archer A is likely to earn 10 points with a probability of $0{\cdot}8$ and Archer B is likely the earn 10 points with a probability of $0{\cdot}9$ . Based on the above information, answer the following questions : If both of them hit the Archery target, then find the probability that
14 (b) $both of them earn 10 points.$
Independent Events
At the start of a cricket match, a coin is tossed and the team winning the toss has the opportunity to choose to bat or bowl. Such a coin is unbiased with equal probabilities of getting head and tail. Based on the above information, answer the following questions :
14 (b) $Find the probability of getting at least one head in three tosses of such a coin.$
Conditional Probability
A shopkeeper sells three types of flower seeds A1, A2, A3. They are sold in the form of a mixture, where the proportions of these seeds are 4 : 4 : 2, respectively. The germination rates of the three types of seeds are 45%, 60% and 35% respectively. Based on the above information :
14 (b) $Calculate the probability that the seed is of type A2, given that a randomly chosen seed germinates.$
Bayes' Theorem

X	0	1	2
P(X)	K	6K	9K

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 2 times: 2023, 2026
18 $P(E) = \dfrac{3}{10}$
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Question 2
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
18 $A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red ?$
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Question 3
Asked 2 times: 2022*, 2024 (* = numbers or wording changed that year)
31 $0{\cdot}3$
Question 4
Asked 2 times: 2022*, 2024 (* = numbers or wording changed that year)
31 (a) $A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.$
Question 5
Asked 2 times: 2023, 2024
30 $A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X.$
Question 6 (case study, 4 parts)
This type asked 5 times: 2022, 2023, 2024, 2025, 2026. This one: 2026
Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information, answer the following questions :
36 (i) $What is the probability that selected person is a female ?$
36 (ii) $If a male person is selected, what is the probability that he will not be suffering from lung problems ?$
36 (iii) (a) $A person selected at random is detected with lung complications. Find the probability that selected person is a female.$
36 (iii) (b) $A person selected at random is not having lung problems, find the probability that the person is a male.$
Question 7
This type asked 5 times: 2022, 2023, 2024, 2025, 2026. This one: 2026
18 $3P(A) = P(B) = \dfrac{3}{5}$
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Question 8
This type asked 5 times: 2022, 2023, 2024, 2025, 2026. This one: 2026
18 $\dfrac{4}{5}$
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Question 9
This type asked 4 times: 2022, 2024, 2025, 2026. This one: 2026
31 (a) $Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.$
Question 10
This type asked 4 times: 2022, 2023, 2025, 2026. This one: 2026
31 (a) $A = \{1,\ 2,\ 5\}$

13.2 Conditional Probability

Solved Examples

Exercise 13.1

13.3 Multiplication Theorem on Probability and 13.4 Independent Events

Solved Examples

Exercise 13.2

13.5 Bayes' Theorem

Solved Examples

Exercise 13.3

Miscellaneous Exercise on Chapter 13

Solved Examples

Miscellaneous Exercise

What the board has asked from this chapter

2026

2025

2024

2023

2022

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6 (case study, 4 parts)

Question 7

Question 8

Question 9

Question 10