Question

If \(P(A) = 0.25\), \(P(B|A) = 0.5\), \(P(B|\bar{A}) = 0.75\) then \(P(A|B)\) is ______.

• A. \(\dfrac{1}{2}\)
• B. \(\dfrac{1}{3}\)
• C. \(\dfrac{3}{8}\)
• D. \(\dfrac{2}{11}\)

Hint:

First find \(P(\bar{A})\) and use the law of total probability to get \(P(B)\), then apply \(P(A|B) = \frac{P(B|A)P(A)}{P(B)}\).

Answer 1

The Correct Option is D

Step 1: Find \(P(\bar{A})\). Since \(P(A) = 0.25\), \[P(\bar{A}) = 1 - 0.25 = 0.75\]

Step 2: Compute \(P(B)\) using the law of total probability: \[P(B) = P(B|A)P(A) + P(B|\bar{A})P(\bar{A})\] \[P(B) = (0.5)(0.25) + (0.75)(0.75) = 0.125 + 0.5625 = 0.6875\]

Step 3: Apply Bayes theorem to get \(P(A|B)\): \[P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{0.125}{0.6875}\]

Step 4: Simplify the fraction. Multiply numerator and denominator by 10000: \[\frac{0.125}{0.6875} = \frac{1250}{6875} = \frac{2}{11}\] (dividing both by 625).

So \(P(A|B) = \dfrac{2}{11}\), which is option (D).