Probability — Mathematics textbook solutions

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 $P(A) = \frac{1}{2}$
1. A. $0$
2. B. $\frac{1}{2}$
3. C. $not defined$
4. D. $1$
Ex 13.1 Q17 $P(A|B) = P(B|A)$
1. A. $A \subset B$
2. B. $A = B$
3. C. $A \cap B = \phi$
4. D. $P(A) = P(B)$

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 $The probability of obtaining an even prime number on each die, when a pair of dice is rolled is$
1. A. $0$
2. B. $\frac{1}{3}$
3. C. $\frac{1}{12}$
4. D. $\frac{1}{36}$
Ex 13.2 Q18 $Two events A and B will be independent, if$
1. A. $A and B are mutually exclusive$
2. B. $P(A'B') = [1 - P(A)]\,[1 - P(B)]$
3. C. $P(A) = P(B)$
4. D. $P(A) + P(B) = 1$

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 $\frac{4}{5}$
1. A. $\frac{4}{5}$
2. B. $\frac{1}{2}$
3. C. $\frac{1}{5}$
4. D. $\frac{2}{5}$
Ex 13.3 Q14 $A \subset B$
1. A. $P(A|B) = \frac{P(B)}{P(A)}$
2. B. $P(A|B) < P(A)$
3. C. $P(A|B) \ge P(A)$
4. D. $None of these$

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 $P(A) \neq 0$
1. A. $A \subset B$
2. B. $B \subset A$
3. C. $B = \phi$
4. D. $A = \phi$
Misc Q12 $P(A|B) > P(A)$
1. A. $P(B|A) < P(B)$
2. B. $P(A \cap B) < P(A)\cdot P(B)$
3. C. $P(B|A) > P(B)$
4. D. $P(B|A) = P(B)$
Misc Q13 $P(A) + P(B) - P(A \text{ and } B) = P(A)$
1. A. $P(B|A) = 1$
2. B. $P(A|B) = 1$
3. C. $P(B|A) = 0$
4. D. $P(A|B) = 0$