If A and B are event such that P(A) = \( \displaystyle\frac{1}{2} \), P(B) = \( p \) and P(A \( \cup \) B) = \( \displaystyle\frac{3}{5} \) then which of following statements is/are TRUE?
(A) Events A and B are independent if \( p=\displaystyle\frac{1}{5} \)
(B) Events A and B are independent if \( p\in(0,1)\sim\left\{\displaystyle\frac{1}{2}\right\} \)
(C) Events A and B are independent then P(A \( \cap \) B) = \( \displaystyle\frac{1}{10} \)
(D) Event A and B are independent then P(A\( \cap \)B) = \( \displaystyle\frac{9}{10} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
By the addition rule:
\( \displaystyle P(A\cap B)=P(A)+P(B)-P(A\cup B) \)
\( \displaystyle =\frac{1}{2}+p-\frac{3}{5} \)
\( \displaystyle =p-\frac{1}{10} \)
A and B are independent exactly when \( P(A\cap B)=P(A)\,P(B)=\frac{p}{2} \):
\( \displaystyle p-\frac{1}{10}=\frac{p}{2} \)
\( \displaystyle \frac{p}{2}=\frac{1}{10} \)
\( \displaystyle p=\frac{1}{5} \)
\( \displaystyle P(A\cap B)=\frac{1}{2}\times\frac{1}{5}=\frac{1}{10} \)
So A and B are independent only when \( p=\frac{1}{5} \), and then \( P(A\cap B)=\frac{1}{10} \).
Checking the statements
(A) Events A and B are independent if \( p=\frac{1}{5} \) — Correct. Independence needs \( p=\frac{1}{5} \).
(B) Events A and B are independent if \( p\in(0,1)\sim\left\{\frac{1}{2}\right\} \) — Incorrect. Independence holds only for \( p=\frac{1}{5} \), not for every other p.
(C) Events A and B are independent then P(A \( \cap \) B) = \( \frac{1}{10} \) — Correct. \( \frac{1}{2}\times\frac{1}{5}=\frac{1}{10} \).
(D) Event A and B are independent then P(A\( \cap \)B) = \( \frac{9}{10} \) — Incorrect. The value is \( \frac{1}{10} \); \( P(A\cap B) \) cannot exceed \( P(A)=\frac{1}{2} \) anyway.
NCERT (2026-27 edition):
Independent Events
