Question

Question 2. To show

If events$\mathit{E}$and$\mathit{F}$are independent, then events $\overline{\mathit{E}}$and $\overline{\mathit{F}}$are also independent.

Short answer

Answer

The events $\overline{E}$and $\overline{F}$are independent.

Step-by-step solution

Step 1. Given information

Events $E$and $F$are independent.

Step 2. Definition and formula to be used

Formula used are:

$P(E\cap F)=P\left(E\right)P\left(F\right)$

Step 3. Evaluate the argument

If events and are independent.

$P(E\cap F)=P\left(E\right)P\left(F\right)$

But, $F=(E\cap F)+(\overline{E}\cup F)$

So,

$P\left(F\right)=P(E\cap F)+P(\overline{E}\cup F)$, which yields

$$\begin{gathered} P(\overline{E}\cap F)=P\left(F\right)-P(E\cap F) \\ =P\left(F\right)-P\left(E\right)P\left(F\right) \\ =P\left(F\right)\left[\right[1-P\left(E\right) \\ =P\left(F\right)P\left(\overline{E}\right) \end{gathered}$$

Repeat the argument for the events$\overline{E}$and $\overline{F}$, this time starting from the statement that are independent $\overline{E}$and $\overline{F}$taking the complement of $F$.

Hence proved.