QUESTION $13$

Consider the following statement: If A and $B$ are independent events of a probability space with sample set $S$ and probability distribution $P$ then A and the complement of $B$ are also independent events.

Complete the following proof by selecting the most appropriate justification for each statement.

We know that the complement of $B,$ namely $\frac{S}{\frac{?}{?}}B,$ is an event by In order to show that this event is indenendent from $A$ we have to show that the probability of their is equal to the product of their probabilities.

$P(A(\frac{S}{\frac{?}{?}}B))=P(\frac{A}{\frac{?}{?}}B)$

$=PA-P(AB)$

$=PA-PA*PB$

$=PA(1-PB)$

$=PA*P(\frac{S}{\frac{?}{?}}B)$

A and $B$ are independent