Probability — Chapter Test | Class 12 IMO

Q1

If A and B are independent events with P(A) = 0.3 and P(A∪B) = 0.6, then P(B) is:

P(A∪B) = P(A) + P(B) − P(A)P(B). So, 0.6 = 0.3 + P(B) − 0.3P(B) → 0.3 = 0.7P(B) → P(B) = 3/7.

Q2

If P(A)=0.5, P(B)=0.4 and P(A∩B)=0.2, then P(B|A) is:

P(B|A)=0.2/0.5=0.4.

Q3

A blood test is 99% accurate in detecting a disease. However, 0.1% of the population actually has the disease. If a person tests positive, the approximate probability they actually have the disease is:

P(Disease|Positive) = (0.001 × 0.99) / [(0.001 × 0.99) + (0.999 × 0.01)] = 0.00099 / 0.01098 ≈ 0.0901 or ~9%.

Q4

An insurance agent sells policies. The expected number of policies sold daily is 2.5. If he earns ₹1000 per policy sold, his expected daily earnings are:

By properties of expectation, E(aX) = aE(X). Here, a = 1000 and E(X) = 2.5. So, Expected earnings = 1000 × 2.5 = ₹2500.

Q5

A bag contains 5 red and 3 blue balls. If a ball drawn is known to be red or blue, the probability it is red is:

Total balls=8. Probability of red=5/8.

Q6

A family has 2 children. Given that at least one of them is a boy, what is the probability that both are boys?

Sample space = {BB, BG, GB, GG}. At least one boy = {BB, BG, GB}. Favourable (both boys) = {BB}. Probability = 1/3.

Q7

A random variable X takes values 1,2,4 with probabilities 0.5,0.3,0.2. Variance is:

E(X)=1.9, E(X²)=4.9. Variance=4.9−3.61=1.29.

Q8

If P(A|B)=0.75 and P(A∩B)=0.15, then P(B) is:

P(B)=0.15/0.75=0.20.

Q9

A card is drawn. Given it is a red card, probability that it is a heart is:

Red cards=26. Hearts=13. Probability=13/26=1/2.

Q10

If P(A)=0.7 and P(B|A)=0.4, then P(A∩B) equals:

P(A∩B)=P(B|A)×P(A)=0.4×0.7=0.28.

Q11

Two dice are thrown. Let X denote the sum of the numbers appearing. The probability P(X=7) is:

The sum 7 can be formed by (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). That is 6 outcomes out of 36. 6/36 = 1/6.

Q12

A random variable X can take any positive integer value x with probability P(X=x) = (1/2)ˣ. The probability that X is an even number is:

P(Even) = P(2) + P(4) + ... = (1/2)² + (1/2)⁴ + ... This is an infinite geometric series with a = 1/4 and r = 1/4. Sum = a/(1−r) = (1/4)/(3/4) = 1/3.

Q13

In a school, 40% of students study Maths, 25% study Physics, and 15% study both. If a student selected at random studies Physics, the probability they also study Maths is:

P(Maths | Physics) = P(Maths ∩ Physics) / P(Physics) = 15% / 25% = 3/5.

Q14

A random variable has probability distribution P(x) = kx² for x = 1, 2, 3. The value of k is:

Sum = k(1²) + k(2²) + k(3²) = k(1 + 4 + 9) = 14k = 1. Therefore, k = 1/14.

Q15

A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope, only two consecutive letters 'TA' are visible. The probability that the letter came from TATANAGAR is:

TATANAGAR has 8 pairs, 'TA' appears twice (prob 2/8 = 1/4). CALCUTTA has 7 pairs, 'TA' appears once (prob 1/7). P(TATA|TA) = (1/2 × 1/4) / [(1/2 × 1/4) + (1/2 × 1/7)] = (1/4) / (1/4 + 1/7) = 7/11.