Probability

Previous Years' CBSE Board Questions

13.1 Introduction

MCQ

1. Five fair coins are tossed simultaneously. The probability of the events that atleast one head comes up is
2. (b) <math>\frac{5}{32}</math> (c) <math>\frac{31}{32}</math> (d) <math>\frac{1}{32}</math>

(2023)

2 A die is thrown once. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then <math>P(A \cup B)</math> is

1. 2 5 (b) 3 (c) 0 (d) 1 (2020)

VSA . (1 mark)

1. A coin is tossed once. If head comes up, a die is thrown, but if tail comes up, the coin is tossed again. Find the probability of obtaining head and number 6. (2021 C)
2. Two cards are drawn at random and one-by-one without replacement from a well-shuffled pack of 52 playing cards. Find the probability that one card is red and the other is black. (2020)
3. From a pack of 52 cards, 3 cards are drawn at random (without replacement). The probability that they are two red cards and one black card, is _____ (2020C)
4. A bag contains 3 black, 4 red and 2 green balls. If three balls are drawn simultaneously at random, then the probability that the balls are of different colours is ______. (2020) Ap

SAI (2 marks)

1. A box B₁ contains 1 white ball and 3 red balls. Another box <math>B_2</math> contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes B₁ and B₂, then find the probability that the two balls drawn are of the same colour. (Term II, 2021-22)

(b) Both (A) and (R) are true, but (R) is not the

correct explanation of the (A)

1. (d) (A) is false, but (R) is true. (2023)

1. A bag contains cards numbered 1 to 25. Two cards are drawn at random, one after the other, without replacement. Find the probability that the number on each card is a multiple of 7.

(Term II, 2021-22 C)

1. If A and B are two events such that <math>P(A) = 0.4</math>, <math>P(B) = 0.3</math> and <math>P(A \cup B) = 0.6</math>, then find <math>P(B' \cap A)</math>. (2020) An

(2020) Ap

Out of 8 outstanding students of a school, in which there are 3 boys and 5 girls, a team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected.

(4 marks)

A bag A contains 4 black and 6 red balls and bag B

contains 7 black and 3 red balls. A die is thrown, If 1 or

2 appears on it, then bag Ais chosen, otherwise bag B. If

two balls are drawn at random (without replacement)

from the selected bag, find the probability of one of

them being red and another black. (Delhi 2015) [cr

13.2 Conditional Probability

MCO

For two events A and B, if P(A) = 0.4, P(B) = 0.8 and

<math>P(B/A) = 0.6</math>, then <math>P(A \cup B)</math> is

1. 0.24 (b) 0.3 (c) 0.48 (d) 0.96 (2023)

If for any two events A and B,

<math>P(A) = \frac{4}{5} \operatorname{and} P(A \cap B) = \frac{7}{10}</math>, then P(B/A) is

(a) <math>\frac{1}{10}</math> (b) <math>\frac{1}{8}</math> (c) <math>\frac{7}{8}</math> (d) 17<br>20

(2023)

14. In the following questions, a statements are

Assertion (A) is followed by a statement of Reason

(R). Choose the correct answer out of the following

choices:

Assertion (A): Two coins are tossed simultaneously.

The probability of getting two heads, if it is known

that at least one head comes up, is <math>\frac{1}{2}</math>.

Reason (R): Let E and F be two events with a random

experiment, then <math>P(F/E) = \frac{P(E \cap F)}{P(F)}</math>.

(a) Both (A) and (R) are true and (R) is the correct

explanation of (A)

A card is picked at random from a pack of 52 playing

cards. Given that the picked card is a queen, the

probability of this card to be a card of spade is (a)

SAI (2 marks)

16. Find <math>[P(B/A) + P(A/B)]</math>, if <math>P(A) = \frac{3}{10}</math>, <math>P(B) = \frac{2}{5}</math> and

<math>P(A \cup B) = \frac{3}{5}</math>. (2020)

(Al 2019)

1. 12 cards numbered 1 to 12 (one number on one card), are placed in a box and mixed up thoroughly. Then a card is drawn at random from the box. If it is known that the number on the drawn card is greater than 5, find the probability that the card bears an odd number. (Al 2019) An

Mother, father and son line up at random for a family photo. If A and B are two events given by <math>A = Son on</math> one end, <math>B = Father in the middle, find <math>P(B/A)</math>.</math>

(2019)

A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

(2018)

LAI (4 marks)

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

1. the youngest is a girl.
2. atleast one is a girl. (Delhi 2014) (Ap)

1. A couple has 2 children. Find the probability that both are boys, if it is known that
2. one of them is a boy,
3. the older child is a boy. (Delhi 2014C)

(5/6 marks)

22 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'. (Delhi 2014C)

13.3 Multiplication Theorem on Probability

(4 marks)

(Delhi 2019) An

1. A bag contains 3 red and 7 black balls. Two balls are selected at random one-by-one without replacement. If the second selected ball happens to be red, what is the probability that the first selected ball is also red?

(Delhi 2014C) (Ev

13.4 Independent Events

MCQ

1. If A and B are two independent events with <math>P(A) = \frac{1}{3}</math> and <math>P(B) = \frac{1}{4}</math>, then <math>P(B'|A)</math> is equal to
2. (b) <math>\frac{1}{3}</math>
3. 3 4 (d) 1 (2020) R

VSA (1 mark)

A problem is given to three students whose probabilities of solving it are <math>\frac{1}{3}</math>, <math>\frac{1}{4}</math> and <math>\frac{1}{6}</math> respectively.

If the events of solving the problem are independent,

find the probability that at least one of them solves it.

(2020) [1]

SAI (2 marks)

1. The probability that A hits the target is <math>\frac{1}{2}</math> and the probability that B hits it, is <math>\frac{2}{5}</math>. If both try to hit the target independently, find the probability that the target is hit. (Term II, 2021-22)
2. Events A and B are such that <math>P(A) = \frac{1}{2}</math>, <math>P(B) = \frac{7}{12}</math> and <math>P(\bar{A} \cup \bar{B}) = \frac{1}{4}</math> Find whether the events A and B are independent or not. (Term II, 2021-22) (Ap)
3. The probability of finding a green signal on a busy crossing X is 30%. What is the probability of finding a green signal on X on two consecutive days out of three? (2020)
4. Given two independent events A and B such that P(A) <math>= 0.3</math> and <math>P(B) = 0.6</math>, find <math>P(A' \cap B')</math>. (2020)

The probability of two students A and B coming to

school on time are <math>\frac{2}{7}</math> and <math>\frac{4}{7}</math> , respectively. Assuming

that the events 'A coming on time' and 'B coming on

time' are independent, find the probability of only

one of them coming to school on time. (2019)

A die marked 1, 2, 3 in red and 4, 5, 6 in green is

tossed. Let A be the event "number is even" and B be

the event "number is marked red". Find whether the

events A and B are independent or not.

OR

A die, whose faces are marked 1, 2, 3 in red and

4, 5, 6 in green, is tossed. Let A be the event "number

obtained is even" and B be the event "number

obtained is red". Find if A and B are independent

events. (Al 2017)

Prove that if E and F are independent events, then

the events E' and F' are also independent. (Delhi 2017)

LA I (4 marks)

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 0.8 and Archer B is likely to earn 10 points with a

probability of 0.9.