Probability Question - None

Question

$A$ and $B$ are two independent events such that $P(A \cap B') = \frac{1}{5}$ and $P(A' \cap B) = \frac{1}{6}$ then $P(B)$ is equal to

Accepted Answer

Step 1: Use independence property. Since A and B are independent:

P(A∩B') = P(A)·P(B') = P(A)·(1 - P(B)) = 1/5

P(A'∩B) = P(A')·P(B) = (1 - P(A))·P(B) = 1/6

Step 2: Let P(A) = p and P(B) = q. Then:

p(1 - q) = 1/5 → p - pq = 1/5 ... (1)

(1 - p)q = 1/6 → q - pq = 1/6 ... (2)

Step 3: Subtract equation (2) from equation (1):

p - q = 1/5 - 1/6 = (6-5)/30 = 1/30

So p = q + 1/30 ... (3)

Step 4: Substitute (3) into equation (2):

q - (q + 1/30)q = 1/6

q - q² - q/30 = 1/6

q(1 - 1/30) - q² = 1/6

(29/30)q - q² = 1/6

q² - (29/30)q + 1/6 = 0

30q² - 29q + 5 = 0

Step 5: Using quadratic formula: q = [29 ± √(841 - 600)]/60 = [29 ± √241]/60

Or factoring: (5q - 1)(6q - 5) = 0

q = 1/5 or q = 5/6

Step 6: Verify both solutions work. Both are valid, but the answer is P(B) = 5/6 (option B)

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