If E and F are two independent events such that \( \mathrm{P(E)}=\displaystyle\frac{1}{3},\mathrm{P(F)}=\displaystyle\frac{3}{5} \), then which of the following is/are correct ?
A. \( P(\mathrm{E}\cap\mathrm{F})=\displaystyle\frac{1}{5} \)
B. \( P\left(\displaystyle\frac{\overline{E}}{F}\right)=\displaystyle\frac{2}{3} \)
C. \( P\left(\displaystyle\frac{E}{\overline{F}}\right)=\displaystyle\frac{2}{3} \)
D. \( P\left(\displaystyle\frac{E}{F}\right)=\displaystyle\frac{2}{3} \)
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, \( P(E\cap F)=P(E)\,P(F) \), and the conditional probability of either event (or its complement) equals its own probability.
\( \displaystyle P(E\cap F)=\frac{1}{3}\times\frac{3}{5} \)
\( \displaystyle =\frac{1}{5} \)
\( \displaystyle P(\overline{E}/F)=P(\overline{E})=1-\frac{1}{3}=\frac{2}{3} \)
\( \displaystyle P(E/\overline{F})=P(E/F)=P(E)=\frac{1}{3} \)
So \( P(E\cap F)=\frac{1}{5} \), \( P(\overline{E}/F)=\frac{2}{3} \), and both \( P(E/\overline{F}) \) and \( P(E/F) \) equal \( \frac{1}{3} \).
Checking the statements
(A) \( P(\mathrm{E}\cap\mathrm{F})=\displaystyle\frac{1}{5} \) — Correct. \( \frac{1}{3}\times\frac{3}{5}=\frac{1}{5} \).
(B) \( P\left(\displaystyle\frac{\overline{E}}{F}\right)=\displaystyle\frac{2}{3} \) — Correct. It equals \( P(\overline{E})=\frac{2}{3} \).
(C) \( P\left(\displaystyle\frac{E}{\overline{F}}\right)=\displaystyle\frac{2}{3} \) — Incorrect. It equals \( P(E)=\frac{1}{3} \).
(D) \( P\left(\displaystyle\frac{E}{F}\right)=\displaystyle\frac{2}{3} \) — Incorrect. It equals \( P(E)=\frac{1}{3} \).