A lottery consists of choosing 11 different numbered balls (the winning numbers) at random from a group of 100 numbers. A player chooses 11 numbers on a ticket. What's the expected number of winning numbers chosen on a ticket?
I tried this method: $$\begin{array}{rcl} && 0\cdot\mathbb{P}(0\text{ winning numbers}) + ....+ 11\cdot \mathbb{P}(11\text{ winning numbers}) \\ & = &0 \cdot\left(\binom{11}{0}/\binom{100}{11}\right) + ....+11\cdot\left(\binom{11}{11}/\binom{100}{11}\right) \end{array}$$
This should be correct but it's a pain to calculate by hand (as I would have a few minutes to do this without a calculator on an exam).
Question: Can anyone think of a faster way?
I'm trying to find a family that represents this process, but can't think of any. Perhaps we can use a new random variable, which is a sum of 1s if the $i$th number matches and 0s if the $i$th number does not match.
[self-study]
tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. $\endgroup$