Expected value with indicator random variables I don't understand the solution for problem 7a at http://www.ma.utexas.edu/users/geir/teaching/m362k/weeklyhw9solns.pdf. 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.


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?
