probability, indicator random variable

Question

Let $A,B,C$ be independent events with $P(A)=P(B)=P(C)=\dfrac{1}{2}$.

Let $X$ be the indicator r.v. of the event $A \cup B$ and $Y$ the indicator r.v. of the event $B \cup C$. Compute ${\bf E}[XY]$.

My attempt:

$E[XY]=P[ (A \cup B) \cap ( B \cup C)]$, from here, it is unclear to me whether $A \cup B$ and $B \cup C$ are independent, so I am stuck.**

Answer 1

You can rewrite $(A\cup B)\cap (C\cup B)=(A\cap C)\cup B$, and then \begin{align*} \mathbb{P}\left((A\cap C)\cup B\right) &= \mathbb{P}\left(A\cap C\right) + \mathbb{P}B - \mathbb{P}\left(A\cap C\cap B\right) \\ &= \mathbb{P} A\cdot \mathbb{P}C + \mathbb{P}B - \mathbb{P} A\cdot \mathbb{P}C\cdot \mathbb{P}B \\ &= \frac{1}{4} + \frac{1}{2} - \frac{1}{8} \\ &= \frac{5}{8}. \end{align*}

Answer 2

Create an indicator variable for $XY$. The possible values of $XY$ are 0 and 1. So $$XY = \begin{cases} 1 \ \ \text{if} \ X \ \text{and} \ Y \ \text{occur} \\ 0 \ \ \text{if} \ \text{only} \ X \ \text{occurs}, \ \text{only} \ Y \ \text{occurs} \ \text{or neither occurs} \end{cases}$$