If A and B are two independent events such that \( P(A)=\displaystyle\frac{4}{7} \) and \( P(B)=\displaystyle\frac{2}{7} \), then \( P(\overline{A}\cap\overline{B}) \) equals
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{15}{49} \)
If A and B are independent, so are their complements \( \overline{A} \) and \( \overline{B} \).
\( \displaystyle P(\overline{A})=1-\frac{4}{7}=\frac{3}{7} \)
\( \displaystyle P(\overline{B})=1-\frac{2}{7}=\frac{5}{7} \)
\( \displaystyle P(\overline{A}\cap\overline{B})=P(\overline{A})\,P(\overline{B}) \)
\( \displaystyle =\frac{3}{7}\times\frac{5}{7}=\frac{15}{49} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{8}{49} \) — Incorrect. \( \frac{8}{49}=P(A)P(B)=P(A\cap B) \), the probability that both events happen.
Option (2) \( \displaystyle\frac{6}{49} \) — Incorrect. \( \frac{6}{49}=\frac{3}{7}\times\frac{2}{7}=P(\overline{A}\cap B) \): the complement of B has not been taken.
Option (3) \( \displaystyle\frac{21}{49} \) — Incorrect. \( \frac{21}{49}=\frac{3}{7}=P(\overline{A}) \) alone.