Question

Question 2. To show

If $\mathit{E}$and$\mathit{F}$ are independent events, prove or disprove that $\overline{\mathit{E}}$and $\mathit{F}$ are necessarily independent events.

Short answer

Answer

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

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 $E$and$F$ 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

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

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

Hence proved.