If \( P(A)=\displaystyle\frac{3}{10} \), \( P(B)=\displaystyle\frac{3}{4} \) and \( P(A\cap B)=\displaystyle\frac{1}{5} \), then \( P(B|A) \) is equal to:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{2}{3} \)
The conditional probability of B given A divides by the probability of the given event A:
\( \displaystyle P(B|A)=\frac{P(A\cap B)}{P(A)} \)
\( \displaystyle =\frac{1/5}{3/10} \)
\( \displaystyle =\frac{1}{5}\times\frac{10}{3} \)
\( \displaystyle =\frac{2}{3} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{4}{15} \) — Incorrect. \( \frac{4}{15}=\frac{1/5}{3/4} \) is \( P(A|B) \), which divides by P(B) instead of P(A).
Option (2) \( \displaystyle\frac{9}{40} \) — Incorrect. \( \frac{9}{40}=\frac{3}{10}\times\frac{3}{4} \) is P(A)P(B), not a conditional probability.
Option (4) \( \displaystyle\frac{3}{5} \) — Incorrect. The value is \( \frac{2}{3} \), not \( \frac{3}{5} \).