Question

Prove that if \(E\) and \(F\) are two independent events, then \(E\) and \(F'\) will also be independent.

Hint:

Split E as (E∩F)∪(E∩F'), use independence of E,F, and the complement rule.

Answer 1

Step 1: Starting from the definition of F':
\(F'\) is the complement of \(F\), so \(P(F')=1-P(F)\), and \(E\cap F'\) consists of all outcomes in \(E\) that are NOT in \(F\).

Step 2: Using the general subtraction rule for sets:
For any two events, \(P(E\cap F')=P(E)-P(E\cap F)\) (since \(E\cap F\) and \(E\cap F'\) are disjoint and together make up all of \(E\)).

Step 3: Substituting the independence hypothesis:
Given \(P(E\cap F)=P(E)P(F)\), substitute: \(P(E\cap F')=P(E)-P(E)P(F)=P(E)(1-P(F))=P(E)P(F')\), which is precisely the definition of independence between \(E\) and \(F'\).

Final Answer:
\[ \boxed{E \text{ and } F' \text{ are independent}} \]