NCERT solutions for Mathematics [English] Class 11 chapter 14 - Probability [Latest edition]

A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive?

A die is thrown. Describe the following events:

1. A: a number less than 7
2. B: a number greater than 7
3. C: a multiple of 3
4. D: a number less than 4
5. E: an even number greater than 4
6. F: a number not less than 3

Also find A ∪ B, A ∩ B, B ∪ C, E ∩ F, D ∩ E, A – C, D – E, E ∩ F', F'

An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

A: the sum is greater than 8, B: 2 occurs on either die

C: The sum is at least 7 and a multiple of 3.

Which pairs of these events are mutually exclusive?

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

1. mutually exclusive?
2. simple?
3. compound?

Three coins are tossed. Describe two events which are mutually exclusive.

Three coins are tossed. Describe three events which are mutually exclusive and exhaustive.

Three coins are tossed. Describe two events, which are not mutually exclusive.

Three coins are tossed. Describe two events which are mutually exclusive but not exhaustive.

Three coins are tossed. Describe 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 ≤ 5

Describe the events

1. A' 
2. not B
3. A or B
4. A and B
5. A but not C
6. B or C
7. B and C
8. A ∩ B' ∩ C'

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 ≤ 5.

State true or false: (give reason for your answer).

A and B are mutually exclusive

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 ≤ 5

State true or false: (give reason for your answer)

A and B are mutually exclusive and 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 ≤ 5

State true or false: (give reason for your answer).

A = B'

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 ≤ 5

State true or false: (give reason for your answer)

 A and C are mutually exclusive

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 ≤ 5

State true or false: (give reason for your answer)

A and B' are mutually exclusive

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 ≤ 5

State true or false: (give reason for your answer)

A' , B' ,C are mutually exclusive and exhaustive.

Which of the following can not be valid assignment of probabilities for outcomes of sample space S = {ω_1, ω_2,ω_3,ω_4,ω_5,ω_6,ω₇}

A die is thrown, find the probability of following events:

1. A prime number will appear,
2. A number greater than or equal to 3 will appear,
3. A number less than or equal to one will appear,
4. A number more than 6 will appear,
5. A number less than 6 will appear.

A card is selected from a pack of 52 cards.

1. How many points are there in the sample space?
2. Calculate the probability that the card is an ace of spades.
3. Calculate the probability that the card is
   
   1. an ace
   2. 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 (i) 3 (ii) 12

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?

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

1. 3 heads
2. 2 heads
3. at least 2 heads
4. at most 2 heads
5. no head
6. 3 tails
7. exactly two tails
8. no tail
9. atmost two tails.

If `2/11` is the probability of an event, what is the probability of the event ‘not A’.

A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is

1. a vowel
2. an consonant

In a lottery, person chooses 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

P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6

Check whether the following probabilities P(A) and P(B) are consistently defined

P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8

Fill in the blank in following table:

Fill in the blank in following table:

Fill in the blank in following table:

Given P(A) = `3/5` and P(B) = `1/5`. Find P(A or B), if A and B are mutually exclusive events.

If E and F are events such that P(E) = `1/4`, P(F) = `1/2` and P(E and F) = `1/8`, find

1. P(E or F)
2. P(not E and not F).

A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine P(not A).

A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine P (not B)

A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine P(A or B).

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.

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 at least one of them is 0.95. What is the probability of passing both?

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

1. The student opted for NCC or NSS.
2. The student has opted neither NCC nor NSS.
3. The student has opted NSS but not NCC.

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 all will be blue?

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 atleast one will be green?

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

1. P(2)
2. P(1 or 3)
3. P(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 one ticket.

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 two tickets.

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 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

1. you both enter the same sections?
2. you both enter the different sections?

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 ∩ B) = 0.35.

Find

1. P(A ∪ B)
2. P(A′ ∩ B′)
3. P(A ∩ B′)
4. P(B ∩ A′)

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:

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, the digits are repeated?

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, the repetition of digits is not allowed?

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?