Independent Event

Problem 114: Roll a Die, and consider the following two events: $A = \{2,4,6\}$ and $B= \{3,6\}$ . Are the events $E$ and $F$ independent?

Solution: If events $A$ and $B$ are independent, then $P(A|B) = P(A)$ and $P(B|A) = P(B)$ .

1. Calculate $P(A)$ :

(1) $\begin{align*} P(A ) = \frac{3}{6} = \frac{1}{2}\end{align*}$

2. Calculate $P(B)$ :

(2) $\begin{align*} P(B) = \frac{2}{6} = \frac{1}{3} \end{align*}$

3. Calculate $P(A|B)$ :

(3) $\begin{align*} P(A|B) & = \frac{P(A \cap B)}{P(B)} \\ & = \frac{1/6}{1/3} \\ & = \frac{3}{6} = \frac{1}{2} \end{align*}$

4. Calculate $P(B|A)$ :

(4) $\begin{align*} P(B|A) & = \frac{P(B \cap A)}{P(A)} \\ & = \frac{1/6}{1/2} \\ & = \frac{2}{6} = \frac{1}{3} \end{align*}$

Since $P(A|B) = P(A)$ and $P(B|A) = P(B)$ , the events $A$ and $B$ are independent.