Question

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

Hint:

Write P(E∩F') = P(E) − P(E∩F) and substitute P(E∩F) = P(E)P(F).

Answer 1

Step 1: Starting from the target identity:
To prove independence of \(E\) and \(F'\), we must show \(P(E\cap F')=P(E)P(F')\).

Step 2: Expressing P(F') and P(E∩F') via complements:
\(P(F')=1-P(F)\), and \(E\cap F'=E\setminus(E\cap F)\), so \(P(E\cap F')=P(E)-P(E\cap F)\).

Step 3: Substituting the independence hypothesis:
Given \(P(E\cap F)=P(E)P(F)\): \(P(E\cap F')=P(E)-P(E)P(F)=P(E)(1-P(F))=P(E)P(F')\).

Final Answer:
This matches the definition of independence for \(E\) and \(F'\).\[ \boxed{P(E\cap F')=P(E)P(F')} \]