I don't understand the solution for problem 7a at . Reproduced below for reference:
Suppose that A and B each randomly, and independently, choose 3 of 10 objects. Find the expected number of objects chosen by both A and B.
solution: Let $X$ be the number of objects chosen by both A and B. For $1 \le i \le 10$, let $$ \begin{align} X_i = \begin{cases} 1 &\mbox{if } \text{object i is chosen by A and B} \\ 0 &\mbox{otherwise } \end{cases} \end{align} $$ Then $X = X_1 + ... + X_{10}$. We find $$ E[X_i] = 0\cdot P(X_i = 0) + 1\cdot P(X_i = 1) = P(X_i = 1) = 9/100. $$ By the linearity of expectation, $$E[X] = 10\cdot E[X_i] = 0.9$$
I don't understand how they reduced $E[X_i]$ (don't they need $P(X_i = 2, 3, ..., 10)$ terms?) and how $9/100$ was computed.
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$\begingroup$I don't understand how they reduced $E[X_i]$ (don't they need $P(X_i = 2, 3, ..., 10)$ terms?)
By definition, the only two values taken by $X_i$ are $0$ and $1$. I think you may be confusing $X_i$ with $i$. You can think of $i$ as being fixed for the duration of the sentence beginning "We find $E[X_i] =$ ..."
and how $9/100$ was computed.
We want the probability that A chose object $i$ and B chose object $i$. These two events are independent by assumption. What does that independence tell us?
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