Mathematics · Textbook solutions

Probability

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 105 questions

14.1 Event

33 q

Solved Examples

Worked · 3

14.1 Eg.1 $A$
14.1 Eg.2 $A$
14.1 Eg.3 $A$

Exercise 14.1

Practice · 30

Ex 14.1 Q1 $E$
A die is thrown. Describe the following events. Also find $A \cup B$ , $A \cap B$ , $B \cup C$ , $E \cap F$ , $D \cap E$ , $A - C$ , $D - E$ , $E \cap F'$ , $F'$ .
Ex 14.1 Q2(i) $A$
Ex 14.1 Q2(ii) $B$
Ex 14.1 Q2(iii) $C$
Ex 14.1 Q2(iv) $D$
Ex 14.1 Q2(v) $E$
Ex 14.1 Q2(vi) $F$
Ex 14.1 Q3 $A$
Three coins are tossed once. Let $A$ denote the event 'three heads show", $B$ denote the event "two heads and one tail show", $C$ denote the event" three tails show and $D$ denote the event 'a head shows on the first coin". Which events are
Ex 14.1 Q4(i) $mutually exclusive?$
Ex 14.1 Q4(ii) $simple?$
Ex 14.1 Q4(iii) $Compound?$
Three coins are tossed. Describe
Ex 14.1 Q5(i) $Two events which are mutually exclusive.$
Ex 14.1 Q5(ii) $Three events which are mutually exclusive and exhaustive.$
Ex 14.1 Q5(iii) $Two events, which are not mutually exclusive.$
Ex 14.1 Q5(iv) $Two events which are mutually exclusive but not exhaustive.$
Ex 14.1 Q5(v) $Three events which are mutually exclusive but not exhaustive.$
Two dice are thrown. The events $A$ , $B$ and $C$ are as follows: $A$ : getting an even number on the first die. $B$ : getting an odd number on the first die. $C$ : getting the sum of the numbers on the dice $\le 5$ . Describe the events
Ex 14.1 Q6(i) $A'$
Ex 14.1 Q6(ii) $B$
Ex 14.1 Q6(iii) $A$
Ex 14.1 Q6(iv) $A$
Ex 14.1 Q6(v) $A$
Ex 14.1 Q6(vi) $B$
Ex 14.1 Q6(vii) $B$
Ex 14.1 Q6(viii) $A \cap B' \cap C'$
Refer to question 6 above, state true or false: (give reason for your answer). [Question 6 sets up: two dice are thrown, with $A$ : getting an even number on the first die; $B$ : getting an odd number on the first die; $C$ : getting the sum of the numbers on the dice $\le 5$ .]
Ex 14.1 Q7(i) $A$
Ex 14.1 Q7(ii) $A$
Ex 14.1 Q7(iii) $A = B'$
Ex 14.1 Q7(iv) $A$
Ex 14.1 Q7(v) $A$
Ex 14.1 Q7(vi) $A'$

14.2 Axiomatic Approach to Probability

48 q

Solved Examples

Worked · 5

14.2 Eg.4

S = \{\omega_1, \omega_2, ..., \omega_6\}

14.2 Eg.5 $One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally likely, calculate the probability that the card will be (i) a diamond (ii) not an ace (iii) a black card (i.e., a club or, a spade) (iv) not a diamond (v) not a black card.$
14.2 Eg.6 $A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are similar in shape and size. A disc is drawn at random from the bag. Calculate the probability that it will be (i) red, (ii) yellow, (iii) blue, (iv) not blue, (v) either red or blue.$
14.2 Eg.7 $Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is 0.05 and that Ashima will qualify the examination is 0.10. The probability that both will qualify the examination is 0.02. Find the probability that (a) Both Anil and Ashima will not qualify the examination. (b) Atleast one of them will not qualify the examination and (c) Only one of them will qualify the examination.$
14.2 Eg.8 $A committee of two persons is selected from two men and two women. What is the probability that the committee will have (a) no man? (b) one man? (c) two men?$

Outcomes	$\omega_1$	$\omega_2$	$\omega_3$	$\omega_4$	$\omega_5$	$\omega_6$
(a)	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$
(b)	$1$	$0$	$0$	$0$	$0$	$0$
(c)	$\frac{1}{8}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$-\frac{1}{4}$	$-\frac{1}{3}$
(d)	$\frac{1}{12}$	$\frac{1}{12}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{3}{2}$
(e)	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$

Exercise 14.2

Practice · 43

Ex 14.2 Q1

S = \{\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}, \omega_{5}, \omega_{6}, \omega_{7}\}

Ex 14.2 Q2 $A coin is tossed twice, what is the probability that atleast one tail occurs?$
A die is thrown, find the probability of following events:
Ex 14.2 Q3(i) $A prime number will appear,$
Ex 14.2 Q3(ii) $A number greater than or equal to 3 will appear,$
Ex 14.2 Q3(iii) $A number less than or equal to one will appear,$
Ex 14.2 Q3(iv) $A number more than 6 will appear,$
Ex 14.2 Q3(v) $A number less than 6 will appear.$
A card is selected from a pack of 52 cards.
Ex 14.2 Q4(a) $How many points are there in the sample space?$
Ex 14.2 Q4(b) $Calculate the probability that the card is an ace of spades.$
Ex 14.2 Q4(c) $Calculate the probability that the card is (i) an ace (ii) black card.$
A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. find the probability that the sum of numbers that turn up is
Ex 14.2 Q5(i) $3$
Ex 14.2 Q5(ii) $12$
Ex 14.2 Q6 $There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?$
Ex 14.2 Q7 $A fair coin is tossed four times, and a person win Re 1 for each head and lose Rs 1.50 for each tail that turns up. From the sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.$
Three coins are tossed once. Find the probability of getting
Ex 14.2 Q8(i) $3 heads$
Ex 14.2 Q8(ii) $2 heads$
Ex 14.2 Q8(iii) $atleast 2 heads$
Ex 14.2 Q8(iv) $atmost 2 heads$
Ex 14.2 Q8(v) $no head$
Ex 14.2 Q8(vi) $3 tails$
Ex 14.2 Q8(vii) $exactly two tails$
Ex 14.2 Q8(viii) $no tail$
Ex 14.2 Q8(ix) $atmost two tails$
Ex 14.2 Q9 $\dfrac{2}{11}$
A letter is chosen at random from the word 'ASSASSINATION'. Find the probability that letter is
Ex 14.2 Q10(i) $a vowel$
Ex 14.2 Q10(ii) $a consonant$
Ex 14.2 Q11 $In a lottery, a person choses six different natural numbers at random from 1 to 20, and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? [Hint order of the numbers is not important.]$
Check whether the following probabilities $P(A)$ and $P(B)$ are consistently defined
Ex 14.2 Q12(i) $P(A) = 0.5$
Ex 14.2 Q12(ii) $P(A) = 0.5$
Ex 14.2 Q13 $P(A)$
Ex 14.2 Q14 $P(A) = \dfrac{3}{5}$
If E and F are events such that $P(E) = \dfrac{1}{4}$ , $P(F) = \dfrac{1}{2}$ and $P(E \text{ and } F) = \dfrac{1}{8}$ , find
Ex 14.2 Q15(i) $P(E \text{ or } F)$
Ex 14.2 Q15(ii) $P(\text{not } E \text{ and not } F)$
Ex 14.2 Q16 $P(\text{not } E \text{ or not } F) = 0.25$
A and B are events such that $P(A) = 0.42$ , $P(B) = 0.48$ and $P(A \text{ and } B) = 0.16$ . Determine
Ex 14.2 Q17(i) $P(\text{not } A)$
Ex 14.2 Q17(ii) $P(\text{not } B)$
Ex 14.2 Q17(iii) $P(A \text{ or } B)$
Ex 14.2 Q18 $In Class XI of a school 40% of the students study Mathematics and 30% study Biology. 10% of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology.$
Ex 14.2 Q19 $In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing atleast one of them is 0.95. What is the probability of passing both?$
Ex 14.2 Q20 $The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of passing the English examination is 0.75, what is the probability of passing the Hindi examination?$
In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that
Ex 14.2 Q21(i) $The student opted for NCC or NSS.$
Ex 14.2 Q21(ii) $The student has opted neither NCC nor NSS.$
Ex 14.2 Q21(iii) $The student has opted NSS but not NCC.$

Assignment	$\omega_{1}$	$\omega_{2}$	$\omega_{3}$	$\omega_{4}$	$\omega_{5}$	$\omega_{6}$	$\omega_{7}$
(a)	0.1	0.01	0.05	0.03	0.01	0.2	0.6
(b)	$\dfrac{1}{7}$	$\dfrac{1}{7}$	$\dfrac{1}{7}$	$\dfrac{1}{7}$	$\dfrac{1}{7}$	$\dfrac{1}{7}$	$\dfrac{1}{7}$
(c)	0.1	0.2	0.3	0.4	0.5	0.6	0.7
(d)	$-0.1$	0.2	0.3	0.4	$-0.2$	0.1	0.3
(e)	$\dfrac{1}{14}$	$\dfrac{2}{14}$	$\dfrac{3}{14}$	$\dfrac{4}{14}$	$\dfrac{5}{14}$	$\dfrac{6}{14}$	$\dfrac{15}{14}$

	$P(A)$	$P(B)$	$P(A \cap B)$	$P(A \cup B)$
(i)	$\dfrac{1}{3}$	$\dfrac{1}{5}$	$\dfrac{1}{15}$	...
(ii)	0.35	...	0.25	0.6
(iii)	0.5	0.35	...	0.7

Miscellaneous Exercise on Chapter 14

24 q

Miscellaneous Examples

Worked · 4

Misc Eg.9 $On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the probability that she visits (i) A before B? (ii) A before B and B before C? (iii) A first and B last? (iv) A either first or second? (v) A just before B?$
Misc Eg.10 $Find the probability that when a hand of 7 cards is drawn from a well shuffled deck of 52 cards, it contains (i) all Kings (ii) 3 Kings (iii) atleast 3 Kings.$
Misc Eg.11 $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$
Misc Eg.12 $In a relay race there are five teams A, B, C, D and E. (a) What is the probability that A, B and C finish first, second and third, respectively. (b) What is the probability that A, B and C are first three to finish (in any order) (Assume that all finishing orders are equally likely)$

Miscellaneous Exercise

Practice · 20

A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn from the box, what is the probability that
Misc Q1(i) $all will be blue?$
Misc Q1(ii) $atleast one will be green?$
Misc Q2 $4 cards are drawn from a well-shuffled deck of 52 cards. What is the probability of obtaining 3 diamonds and one spade?$
A die has two faces each with number '1', three faces each with number '2' and one face with number '3'. If die is rolled once, determine
Misc Q3(i) $P(2)$
Misc Q3(ii) $P(1 \text{ or } 3)$
Misc Q3(iii) $P(\text{not } 3)$
In a certain lottery 10,000 tickets are sold and ten equal prizes are awarded. What is the probability of not getting a prize if you buy
Misc Q4(a) $one ticket$
Misc Q4(b) $two tickets$
Misc Q4(c) $10 tickets.$
Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that
Misc Q5(a) $you both enter the same section?$
Misc Q5(b) $you both enter the different sections?$
Misc Q6 $Three letters are dictated to three persons and an envelope is addressed to each of them, the letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.$
A and B are two events such that $P(A) = 0.54$ , $P(B) = 0.69$ and $P(A \cap B) = 0.35$ . Find
Misc Q7(i) $P(A \cup B)$
Misc Q7(ii) $P(A' \cap B')$
Misc Q7(iii) $P(A \cap B')$
Misc Q7(iv) $P(B \cap A')$
Misc Q8 $From the employees of a company, 5 persons are selected to represent them in the managing committee of the company. Particulars of five persons are as follows: S. No. Name Sex Age in years 1. Harish M 30 2. Rohan M 33 3. Sheetal F 46 4. Alis F 28 5. Salim M 41 A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over 35 years?$
If 4-digit numbers greater than 5,000 are randomly formed from the digits 0, 1, 3, 5, and 7, what is the probability of forming a number divisible by 5 when,
Misc Q9(i) $the digits are repeated?$
Misc Q9(ii) $the repetition of digits is not allowed?$
Misc Q10 $The number lock of a suitcase has 4 wheels, each labelled with ten digits i.e., from 0 to 9. The lock opens with a sequence of four digits with no repeats. What is the probability of a person getting the right sequence to open the suitcase?$

S. No.	Name	Sex	Age in years
1.	Harish	M	30
2.	Rohan	M	33
3.	Sheetal	F	46
4.	Alis	F	28
5.	Salim	M	41