If A and B are two independent events such that P(A) = 1/3, \( P(A\cup B) \) = 3/5 and P(B) = p, then which of the following are true?
(A) The value of p is \( \displaystyle\frac{2}{5} \)
(B) P (A \( \cap \) B) = \( \displaystyle\frac{2}{15} \)
(C) P (B/A) = \( \displaystyle\frac{1}{3} \)
(D) P ( A/B) = \( \displaystyle\frac{3}{5} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
For independent events, \( P(A\cap B)=P(A)\,P(B)=\frac{p}{3} \). By the addition rule:
\( \displaystyle P(A\cup B)=\frac{1}{3}+p-\frac{p}{3}=\frac{1}{3}+\frac{2p}{3} \)
\( \displaystyle \frac{1}{3}+\frac{2p}{3}=\frac{3}{5} \)
\( \displaystyle \frac{2p}{3}=\frac{4}{15} \)
\( \displaystyle p=\frac{2}{5} \)
\( \displaystyle P(A\cap B)=\frac{1}{3}\times\frac{2}{5}=\frac{2}{15} \)
For independent events the condition does not change the probability.
So p = \( \frac{2}{5} \), P(A \( \cap \) B) = \( \frac{2}{15} \), P(B/A) = P(B) = \( \frac{2}{5} \) and P(A/B) = P(A) = \( \frac{1}{3} \).
Checking the statements
(A) The value of p is \( \frac{2}{5} \) — Correct. p = \( \frac{2}{5} \).
(B) P (A \( \cap \) B) = \( \frac{2}{15} \) — Correct. \( \frac{1}{3}\times\frac{2}{5}=\frac{2}{15} \).
(C) P (B/A) = \( \frac{1}{3} \) — Incorrect. P(B/A) = P(B) = \( \frac{2}{5} \).
(D) P ( A/B) = \( \frac{3}{5} \) — Incorrect. P(A/B) = P(A) = \( \frac{1}{3} \).