Question

If E and F are independent events then:

• A. E & F mutually exclusive
• B. E & $\overline{F}$ (complement of the event F) are independent
• C. $\overline{E}$ & $\overline{F}$ are independent
• D. $P\left(E/F\right)+P\left(\overline{E}/F\right)=1$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $\overline{E}$ & $\overline{F}$ are independent
C E & $\overline{F}$ (complement of the event F) are independent
D $P\left(E/F\right)+P\left(\overline{E}/F\right)=1$

Since events E and F are independent, $P(E\cap F)=P\left(E\right).P\left(F\right)$
Also their complements will be independent with each other.
So, $E,\overline{F}$ and $\overline{E},F$ and $\overline{E},\overline{F}$ are also independent.
Also, in the fourth option, if $P\left(F\right)$ is multiplied on both sides of the equation, we have $P(E\cap F).P(\overline{E}\cap F)=P\left(F\right)$ and thus, $P\left(\frac{E}{F}\right)+P\left(\frac{\overline{E}}{F}\right)=1$, which is obviously true!