Conditional Probability Questions

Conditional probability questions with solutions are given here for students to practice and understand the concept of conditional probability. By conditional probability, we mean the possibility of happening an event after the occurrence of an event already. For example, a dice is rolled, and the outcome is an odd number, now we have to find the probability of the outcome being a prime number, then,

A ≡ Odd outcome for rolling a dice

B ≡ Outcome being a prime number

Clearly, A already happened thereafter; we have to find the probability of happening B.

Thus, P(B|A) = P(A ∩ B)/P(A), where P(A) ≠ 0 and

P(B|A) = probability of occurrence of B given A already happened

P(A ∩ B) = probability of occurrence of A and B together

P(A) = probability of occurrence of A.

Learn more about conditional probability.

Conditional Probability Questions with Solutions

Let us solve some questions based on conditional probability with detailed solutions.

Question 1:

Ten numbered cards are there from 1 to 15, and two cards a chosen at random such that the sum of the numbers on both the cards is even. Find the probability that the chosen cards are odd-numbered.

Solution:

Let, A ≡ event of selecting two odd-numbered cards

B ≡ event of selecting cards whose sum is even.

Then,

n(B) = number of ways of choosing two numbers whose sum is even = ⁸C₂ + ⁷C₂.

n(A ∩ B) = number of ways of choosing odd-numbered cards such that their sum is even.

= ⁸C₂.

Now, P(A|B) = P(A ∩ B)/P(B) = n(A ∩ B)/n(B)

= ⁸C₂ / (⁸C₂ + ⁷C₂) = 4/7.

Question 2:

Let E and F are events of a experiment such that P(E) = 3/10 P(F) = ½ and P(F|E) = ⅖. Find the value of (i) P(E ∩ F) (ii) P(E|F) (iii) P(E U F)

Solution:

We know that P(A|B) = P(A ∩ B)/P(B) ⇒ P(A ∩ B) = P(A|B).P(B)

∴ P(E ∩ F) = P(F|E).P(E) =⅖ × 3/10 = 3/25

(ii) P(E|F) = P(E ∩ F)/P(F) = (3/25) ÷ (½) = 6/25

(iii) P(E U F) = P(E) + P(F) – P(E ∩ F) = 3/10 + ½ – 3/25 = 17/25.

Question 3:

The probability of a student passing in science is ⅘ and the of the student passing in both science and maths is ½. What is the probability of that student passing in maths knowing that he passed in science?

Solution:

Let A ≡ event of passing in science

B ≡ event of passing in maths

Given, P(B) = ⅘ and P(A ∩ B) = ½

Then, probability of passing maths after passing in science = P(B|A) = P(A ∩ B)/P(A)

= ½ ÷ ⅘ = ⅝

∴ the probability of passing in maths is ⅝.

Question 4:

In a survey among few people, 60% read Hindi newspaper, 40% read English newspaper and 20% read both. If a person is chosen at random and if he already reads English newspaper find the probability that he also reads Hindi newspaper.

Solution:

Let there be 100 people in the survey, then

Number of people read Hindi newspaper = n(A) = 60

Number of people read English newspaper = n(B) = 40

Number of people read both = n(A ∩ B) = 20

Probability of the person reading Hindi newspaper when he already reads English newspaper is given by –

P(A|B) = n(A ∩ B)/n(B) = 20/40 = ½.

Also Read:

• Bayes Theorem
• Probability for Class 12
• Basic Probability Formulas
• Mutually Exclusive Events

Question 5:

A fair coin is tossed twice such that E: event of having both head and tail, and F: event of having atmost one tail. Find P(E), P(F) and P(E|F)

Solution:

The sample space S = { HH, HT, TH, TT}

E = {HT, TH}

F = {HH, HT, TH}

E ∩ F = {HT, TH}

P(E) = 2/4 = ½

P(F) = ¾

P(E ∩ F) = 2/4 = ½

P(E|F) = P(E ∩ F)/P(F) = ½ ÷ ¾ = ⅔.

Question 6:

In a class, 40% of the students like Mathematics and 25% of students like Physics and 15% like both the subjects. One student select at random, find the probability that he likes Physics if it is known that he likes Mathematics.

Solution:

Let there be 100 students, then,

Number of students like Mathematics = n(A) = 40

Number of students like Physics = n(B) = 25

Number of students like both Mathematics and Physics = n(A ∩ B) = 15

Now, the probability that the student likes Physics if it is known that he likes Mathematics is given by –

P(B|A) = n(A ∩ B)/n(A) = 15/40 = ⅜.

Question 7:

Two dice are rolled, if it is known that atleast one of the dice always shows 4, find the probability that the numbers appeared on the dice have a sum 8.

Solution:

Let,

A: one of the outcomes is always 4

B: sum of the outcomes is 8

Then, A = {(1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4), (4, 1), (4, 2), (4, 3), (4, 5), (4, 6)}

B{(4, 4), (5, 3), (3, 5), (6, 2), (2, 6)}

n(A) = 11, n(B) = 5, n(A ∩ B) = 1

P(B|A) = n(A ∩ B)/n(A) = 1/11.

Question 8:

A bag contains 3 red and 7 black balls. Two balls are drawn at randon without replacement. If the second ball is red, what is the probability that the first ball is also red?

Solution:

Let A: event of selecting a red ball in first draw

B: event of selecting a red ball in second draw

P(A ∩ B) = P(selecting both red balls) = 3/10 × 2/9 = 1/15

P(B) = P(selecting a red ball in second draw) = P(red ball and rad ball or black ball and red ball)

= P(red ball and red ball) + P(black ball and red ball)

= 3/10 × 2/9 + 7/10 × 3/9 = 3/10

∴ P(A|B) = P(A ∩ B)/P(B) = 1/15 ÷ 3/10 = 2/9.

Question 9:

If a family has two children, what is the conditional probability that both are girls if there is atleast one girl?

Solution:

Let A: both being girls

B: Atleast one girl

n(A) = 1

n(B) = 3

n(A ∩ B) = 1

P(A|B) = n(A ∩ B)/n(B) = ⅓.

Question 10:

A dice and a coin are tossed simultaneously. Find the probability of obtaining a 6, given that a head came up.

Solution:

Let A: six coming with a heads

B: coin shows a head

A ={(6, H))

B = {(1, H), (2, H), (3, H), (4, H), (5, H), (6, H)}

n(A ∩ B) = 1 and n(B) = 6

∴ Probability of getting a six when there is a head is given by –

P(A|B) = n(A ∩ B)/n(B) = ⅙.

Practice Questions on Conditional Probability

1. If P(A) = 2P(B) = 4/13 and P(A|B) = ⅔, find the value of P(A U B).

2. In a survey among few people, 60% read Hindi newspaper, 40% read English newspaper and 20% read both. If a person is chosen at random and if he already reads Hindi newspaper find the probability that he also reads English newspaper.

3. A fair coin is tossed twice such that E: event of having both head and tail, and F: event of having atmost one head. Find P(E ∩ F) and P(F|E).

4. In a class, 60% of the students like Mathematics and 35% of students like Physics and 25% like both the subjects. One student select at random, find the probability that he likes Physics if it is known that he likes Mathematics.

5. Two dice are rolled, if it is known that the second dice always shows 4, find the probability that the numbers appeared on the dice have a sum 6.

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