Question

A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of occurrence of A.
1. 1/2
2. 1/3
3. Both A and B
4. None of the Above

Answer 1

Correct Answer - Option 3 : Both A and B

Calculation:

It is given that P (A).P (B) = 1/6

⇒ And also P (A^') P (B^') = 1/3

⇒ This can also be written as [1 - P (A)]. [1 - P (B)] = 1/3

⇒ Let P(A) = x and P(B) = y

⇒ Then (1 - x)(1 - y) = 1/3 and xy = 1/6

⇒ Hence, 1 - x - y + xy = 1/3 and xy = 1/6

⇒ x + y = 5/6 and xy = 1/6

⇒ x(5/6 – x) = 1/6

⇒ 6x² – 5x + 1 = 0.

⇒ (3x - 1) (2x - 1) = 0

⇒ So x = 1/3 and 1/2

⇒ Hence, the probability of A i.e. P(A) = 1/3 or 1/2

⇒ The number of outcomes favourable to A are denoted by n(A) and total number of outcomes in sample space are denoted by n(S). Hence, the formula becomes P(A) = n(A)/n(S).

⇒ The probability of an event can vary between 0 to 1, i.e. 0 ≤ p ≤ 1.

⇒ Probability can never be negative.

⇒ Probability of occurrence of an event = 1 – (Probability that it doesn’t occur).

⇒ P(A∩B) = P(A) × P(B|A) ; if P(A) ≠ 0

⇒ P(A∩B) = P(B) × P(A|B) ; if P(B) ≠ 0