Question

If \(A\) and \(B\) are independent random variables, then which of the following is correct?

• A. \(P(A^c\mid B)=P(A^c)\)
• B. \(P(A\mid B)=P(B)\)
• C. \(P(A\cup B)=P(A)\,P(B)\)
• D. \(P(A^c\cup B^c)=P(A^c)+P(B^c)\)

Hint:

If \(A\) and \(B\) are independent, \[ \boxed{ P(A\mid B)=P(A),\qquad P(A^c\mid B)=P(A^c). } \]

Answer 1

The Correct Option is A

Step 1: Recall what independence gives us directly.
If $A$ and $B$ are independent, then $P(A\mid B)=P(A)$, since knowing $B$ tells us nothing about $A$.
Step 2: Move to the complement using the conditional split.
Since $P(A\mid B)+P(A^c\mid B)=1$ (these are complementary events given $B$), we get $P(A^c\mid B)=1-P(A\mid B)=1-P(A)$.
Step 3: Simplify the right side.
But $1-P(A)=P(A^c)$ by definition of the complement. So $P(A^c\mid B)=P(A^c)$.
\[ \boxed{P(A^c\mid B)=P(A^c)} \]