If A and B are two independent events, then which of the following statements are TRUE ?
(A) \( \mathrm{P}(\mathrm{A}\cup\mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A}).\mathrm{P}(\mathrm{B}) \)
(B) \( \mathrm{P}(\mathrm{A}\cup\mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) \)
(C) \( P(\text{neither A nor B})=P(\bar{A}).P(\bar{B}) \)
(D) \( \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=1 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
For independent events,
\( \displaystyle P(A\cap B)=P(A)P(B) \)
and the complements \( \bar{A} \) and \( \bar{B} \) are also independent, so
\( \displaystyle P(\bar{A}\cap\bar{B})=P(\bar{A})P(\bar{B}) \)
So \( P(A\cap B)=P(A)P(B) \) and \( P(\bar{A}\cap\bar{B})=P(\bar{A})P(\bar{B}) \).
Checking the statements
(A) \( \mathrm{P}(\mathrm{A}\cup\mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A}).\mathrm{P}(\mathrm{B}) \) — Correct. Put \( P(A\cap B)=P(A)P(B) \) in \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \).
(B) \( \mathrm{P}(\mathrm{A}\cup\mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B}) \) — Incorrect. It would need \( P(A)P(B)=0 \), which is false for independent events with non-zero probabilities.
(C) \( P(\text{neither A nor B})=P(\bar{A}).P(\bar{B}) \) — Correct. “Neither A nor B” is \( \bar{A}\cap\bar{B} \), and \( \bar{A} \), \( \bar{B} \) are independent.
(D) \( \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=1 \) — Incorrect. Independence puts no condition on the sum; for example, two independent events can have P(A) = P(B) = 0.2, with sum 0.4.