NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 16 - Probability [Latest edition]

An ordinary deck of cards contains 52 cards divided into four suits. The red suits are diamonds and hearts and black suits are clubs and spades. The cards J, Q, and K are called face cards. Suppose we pick one card from the deck at random. What is the sample space of the experiment?

An ordinary deck of cards contains 52 cards divided into four suits. The red suits are diamonds and hearts and black suits are clubs and spades. The cards J, Q, and K are called face cards. Suppose we pick one card from the deck at random. What is the event that the chosen card is a black face card?

Suppose that each child born is equally likely to be a boy or a girl. Consider a family with exactly three children. List the eight elements in the sample space whose outcomes are all possible genders of the three children.

Suppose that each child born is equally likely to be a boy or a girl. Consider a family with exactly three children. 
Write the following events as a set and find its probability:
The event that exactly one child is a girl.

Suppose that each child born is equally likely to be a boy or a girl. Consider a family with exactly three children. 
Write the following events as a set and find its probability:
The event that at least two children are girls

Suppose that each child born is equally likely to be a boy or a girl. Consider a family with exactly three children. 
Write the following events as a set and find its probability:
The event that no child is a girl

How many two-digit positive integers are multiples of 3?

What is the probability that a randomly chosen two-digit positive integer is a multiple of 3?

A typical PIN (personal identification number) is a sequence of any four symbols chosen from the 26 letters in the alphabet and the ten digits. If all PINs are equally likely, what is the probability that a randomly chosen PIN contains a repeated symbol?

An experiment has four possible outcomes A, B, C and D, that are mutually exclusive. Explain why the following assignments of probabilities are not permissible:

P(A) = 0.12, P(B) = 0.63, P(C) = 0.45, P(D) = – 0.20

An experiment has four possible outcomes A, B, C and D, that are mutually exclusive. Explain why the following assignments of probabilities are not permissible:

P(A) = `9/120`, P(B) = `45/120`, P(C) = `27/120`, P(D) = `46/120`

Probability that a truck stopped at a roadblock will have faulty brakes or badly worn tires are 0.23 and 0.24, respectively. Also, the probability is 0.38 that a truck stopped at the roadblock will have faulty brakes and/or badly working tires. What is the probability that a truck stopped at this roadblock will have faulty breaks as well as badly worn tires?

If a person visits his dentist, suppose the probability that he will have his teeth cleaned is 0.48, the probability that he will have a cavity filled is 0.25, the probability that he will have a tooth extracted is 0.20, the probability that he will have a teeth cleaned and a cavity filled is 0.09, the probability that he will have his teeth cleaned and a tooth extracted is 0.12, the probability that he will have a cavity filled and a tooth extracted is 0.07, and the probability that he will have his teeth cleaned, a cavity filled, and a tooth extracted is 0.03. What is the probability that a person visiting his dentist will have atleast one of these things done to him?

An urn contains twenty white slips of paper numbered from 1 through 20, ten red slips of paper numbered from 1 through 10, forty yellow slips of paper numbered from 1 through 40, and ten blue slips of paper numbered from 1 through 10. If these 80 slips of paper are thoroughly shuffled so that each slip has the same probability of being drawn. Find the probabilities of drawing a slip of paper that is blue or white

An urn contains twenty white slips of paper numbered from 1 through 20, ten red slips of paper numbered from 1 through 10, forty yellow slips of paper numbered from 1 through 40, and ten blue slips of paper numbered from 1 through 10. If these 80 slips of paper are thoroughly shuffled so that each slip has the same probability of being drawn. Find the probabilities of drawing a slip of paper that is numbered 1, 2, 3, 4 or 5

An urn contains twenty white slips of paper numbered from 1 through 20, ten red slips of paper numbered from 1 through 10, forty yellow slips of paper numbered from 1 through 40, and ten blue slips of paper numbered from 1 through 10. If these 80 slips of paper are thoroughly shuffled so that each slip has the same probability of being drawn. Find the probabilities of drawing a slip of paper that is red or yellow and numbered 1, 2, 3 or 4

An urn contains twenty white slips of paper numbered from 1 through 20, ten red slips of paper numbered from 1 through 10, forty yellow slips of paper numbered from 1 through 40, and ten blue slips of paper numbered from 1 through 10. If these 80 slips of paper are thoroughly shuffled so that each slip has the same probability of being drawn. Find the probabilities of drawing a slip of paper that is numbered 5, 15, 25, or 35

An urn contains twenty white slips of paper numbered from 1 through 20, ten red slips of paper numbered from 1 through 10, forty yellow slips of paper numbered from 1 through 40, and ten blue slips of paper numbered from 1 through 10. If these 80 slips of paper are thoroughly shuffled so that each slip has the same probability of being drawn. Find the probabilities of drawing a slip of paper that is white and numbered higher than 12 or yellow and numbered higher than 26.

In a leap year the probability of having 53 Sundays or 53 Mondays is ______.

Three-digit numbers are formed using the digits 0, 2, 4, 6, 8. A number is chosen at random out of these numbers. What is the probability that this number has the same digits?

Three squares of chessboard are selected at random. The probability of getting 2 squares of one colour and other of a different colour is ______.

If A and B are any two events having P(A ∪ B) = `1/2` and P`(barA) = 2/3`, then the probability of `barA ∩ B` is ______.

Three of the six vertices of a regular hexagon are chosen at random. What is the probability that the triangle with these vertices is equilateral?

If A, B, C are three mutually exclusive and exhaustive events of an experiment such that 3P(A) = 2P(B) = P(C), then P(A) is equal to ______.

One mapping (function) is selected at random from all the mappings of the set A = {1, 2, 3, ..., n} into itself. The probability that the mapping selected is one to one is ______.

If the letters of the word ALGORITHM are arranged at random in a row what is the probability the letters GOR must remain together as a unit?

Six new employees, two of whom are married to each other, are to be assigned six desks that are lined up in a row. If the assignment of employees to desks is made randomly, what is the probability that the married couple will have nonadjacent desks?

Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9.

An experiment consists of rolling a die until a 2 appears. How many elements of the sample space correspond to the event that the 2 appears on the kth roll of the die?

An experiment consists of rolling a die until a 2 appears. How many elements of the sample space correspond to the event that the 2 appears not later than the k th roll of the die?

A die is loaded in such a way that each odd number is twice as likely to occur as each even number. Find P(G), where G is the event that a number greater than 3 occurs on a single roll of the die.

In a large metropolitan area, the probabilities are 0.87, 0.36, 0.30 that a family (randomly chosen for a sample survey) owns a colour television set, a black and white television set, or both kinds of sets. What is the probability that a family owns either anyone or both kinds of sets?

If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(A′)

If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(B′)

If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(A ∪ B)

If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(A ∩ B)

If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(A ∩ B′)

If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find P(A′ ∩ B′)

A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, 0.15, 0.20, 0.31, 0.26, .08. Find the probabilities that a particular surgery will be rated complex or very complex 

A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, 0.15, 0.20, 0.31, 0.26, .08. Find the probabilities that a particular surgery will be rated neither very complex nor very simple 

A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, 0.15, 0.20, 0.31, 0.26, .08. Find the probabilities that a particular surgery will be rated routine or complex

A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, 0.15, 0.20, 0.31, 0.26, .08. Find the probabilities that a particular surgery will be rated routine or simple

Four candidates A, B, C, D have applied for the assignment to coach a school cricket team. If A is twice as likely to be selected as B, and B and C are given about the same chance of being selected, while C is twice as likely to be selected as D, what are the probabilities that C will be selected?

Four candidates A, B, C, D have applied for the assignment to coach a school cricket team. If A is twice as likely to be selected as B, and B and C are given about the same chance of being selected, while C is twice as likely to be selected as D, what are the probabilities that A will not be selected?

One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently the sample space consists of four elementary outcomes S = {John promoted, Rita promoted, Aslam promoted, Gurpreet promoted} You are told that the chances of John’s promotion is same as that of Gurpreet, Rita’s chances of promotion are twice as likely as Johns. Aslam’s chances are four times that of John.
Determine P(John promoted)
P(Rita promoted)
P(Aslam promoted)
P(Gurpreet promoted)

One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently the sample space consists of four elementary outcomes S = {John promoted, Rita promoted, Aslam promoted, Gurpreet promoted} You are told that the chances of John’s promotion is same as that of Gurpreet, Rita’s chances of promotion are twice as likely as Johns. Aslam’s chances are four times that of John. If A = {John promoted or Gurpreet promoted}, find P(A).

The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P(A ∩ B) = .07). Determine P(A)

The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P(A ∩ B) = .07). Determine `P(B ∩ barC)`

The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P(A ∩ B) = .07). Determine P(A ∪ B)

The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P(A ∩ B) = .07). Determine `P(A ∩ barB)`

The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P(A ∩ B) = .07). Determine P(B ∩ C)

The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P(A ∩ B) = .07). Determine Probability of exactly one of the three occurs

One urn contains two black balls (labelled B₁ and B₂) and one white ball. A second urn contains one black ball and two white balls (labelled W₁ and W₂). Suppose the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball. Write the sample space showing all possible outcomes

One urn contains two black balls (labelled B₁ and B₂) and one white ball. A second urn contains one black ball and two white balls (labelled W₁ and W₂). Suppose the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball. What is the probability that two black balls are chosen?

One urn contains two black balls (labelled B₁ and B₂) and one white ball. A second urn contains one black ball and two white balls (labelled W₁ and W₂). Suppose the following experiment is performed. One of the two urns is choosen at random. Next a ball is randomly chosen from the urn. Then a second ball is choosen at random from the same urn without replacing the first ball. What is the probability that two balls of opposite colour are choosen?

A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that all the three balls are white

A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that all the three balls are red

A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that all one ball is red and two balls are white

If the letters of the word ASSASSINATION are arranged at random. Find the probability that four S’s come consecutively in the word

If the letters of the word ASSASSINATION are arranged at random. Find the probability that two I’s and two N’s come together

If the letters of the word ASSASSINATION are arranged at random. Find the probability that all A’s are not coming together

If the letters of the word ASSASSINATION are arranged at random. Find the probability that no two A’s are coming together

A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.

A sample space consists of 9 elementary outcomes e₁, e₂, ..., e₉ whose probabilities are 
P(e₁) = P(e₂) = 0.08, P(e₃) = P(e₄) = P(e₅) = 0.1
P(e₆) = P(e₇) = 0.2, P(e₈) = P(e₉) = 0.07
Suppose A = {e₁, e₅, e₈}, B = {e₂, e₅, e₈, e₉}
Calculate P(A), P(B), and P(A ∩ B)

A sample space consists of 9 elementary outcomes e₁, e₂, ..., e₉ whose probabilities are 
P(e₁) = P(e₂) = 0.08, P(e₃) = P(e₄) = P(e₅) = 0.1
P(e₆) = P(e₇) = 0.2, P(e₈) = P(e₉) = 0.07
Suppose A = {e₁, e₅, e₈}, B = {e₂, e₅, e₈, e₉}
Using the addition law of probability, calculate P(A ∪ B)

A sample space consists of 9 elementary outcomes e₁, e₂, ..., e₉ whose probabilities are 
P(e₁) = P(e₂) = 0.08, P(e₃) = P(e₄) = P(e₅) = 0.1
P(e₆) = P(e₇) = 0.2, P(e₈) = P(e₉) = 0.07
Suppose A = {e₁, e₅, e₈}, B = {e₂, e₅, e₈, e₉}
List the composition of the event A ∪ B, and calculate P(A ∪ B) by adding the probabilities of the elementary outcomes.

A sample space consists of 9 elementary outcomes e₁, e₂, ..., e₉ whose probabilities are 
P(e₁) = P(e₂) = 0.08, P(e₃) = P(e₄) = P(e₅) = 0.1
P(e₆) = P(e₇) = 0.2, P(e₈) = P(e₉) = 0.07
Suppose A = {e₁, e₅, e₈}, B = {e₂, e₅, e₈, e₉}
Calculate `P(barB)` from P (B), also calculate `P(barB)` directly from the elementary outcomes of `barB`

Determine the probability p, for the following events.
An odd number appears in a single toss of a fair die.

Determine the probability p, for the following events. 
At least one head appears in two tosses of a fair coin.

Determine the probability p, for the following events. 
A king, 9 of hearts, or 3 of spades appears in drawing a single card from a well-shuffled ordinary deck of 52 cards.

Determine the probability p, for the following events. 
The sum of 6 appears in a single toss of a pair of fair dice.

In a non-leap year, the probability of having 53 tuesdays or 53 wednesdays is ______.

Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive ______.

While shuffling a pack of 52 playing cards, 2 are accidentally dropped. Find the probability that the missing cards to be of different colours ______.

Seven persons are to be seated in a row. The probability that two particular persons sit next to each other is ______.

Without repetition of the numbers, four-digit numbers are formed with the numbers 0, 2, 3, 5. The probability of such a number divisible by 5 is ______.

If A and B are mutually exclusive events, then ______.

If P(A ∪ B) = P(A ∩ B) for any two events A and B, then ______.

6 boys and 6 girls sit in a row at random. The probability that all the girls sit together is ______.

A single letter is selected at random from the word ‘PROBABILITY’. The probability that it is a vowel is ______.

If the probabilities for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that either A or B fails is ______.

The probability that at least one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2, then `P(barA) + P(barB)` is ______.

If M and N are any two events, the probability that at least one of them occurs is ______. 

The probability that a person visiting a zoo will see the giraffee is 0.72, the probability that he will see the bears is 0.84 and the probability that he will see both is 0.52.

The probability that a student will pass his examination is 0.73, the probability of the student getting a compartment is 0.13, and the probability that the student will either pass or get compartment is 0.96.

The probabilities that a typist will make 0, 1, 2, 3, 4, 5 or more mistakes in typing a report are, respectively, 0.12, 0.25, 0.36, 0.14, 0.08, 0.11. 

If A and B are two candidates seeking admission in an engineering College. The probability that A is selected is .5 and the probability that both A and B are selected is at most .3. Is it possible that the probability of B getting selected is 0.7?

The probability of intersection of two events A and B is always less than or equal to those favourable to the event A.

The probability of an occurrence of event A is 0.7 and that of the occurrence of event B is 0.3 and the probability of occurrence of both is 0.4

The sum of probabilities of two students getting distinction in their final examinations is 1.2

The probability that the home team will win an upcoming football game is 0.77, the probability that it will tie the game is 0.08, and the probability that it will lose the game is ______.

If e₁, e₂, e₃, e₄ are the four elementary outcomes in a sample space and P(e1) = 0.1, P(e₂) = 0.5, P(e₃) = 0.1, then the probability of e₄ is ______.

Let S = {1, 2, 3, 4, 5, 6} and E = {1, 3, 5}, then `barE` is ______.

If A and B are two events associated with a random experiment such that P(A) = 0.3, P(B) = 0.2 and P(A ∩ B) = 0.1, then the value of `P(A ∩ barB)` is ______.

The probability of happening of an event A is 0.5 and that of B is 0.3. If A and B are mutually exclusive events, then the probability of neither A nor B is ______.