In the previous chapters, I have several times talked about what happens “on average” or what you can “expect” in situations where there is randomness involved. For example, on page 95, it was pointed out that the parameter λ in the Poisson distribution is the *average* number of occurrences. I have mentioned that each roulette number shows up on *average* once every 38 times and that you can *expect* two sixes if you roll a die 12 times. The time has come to make this discussion exact, to look beyond probabilities and introduce what probabilists call the *expected value* or *mean* This single number summarizes an experiment, and in order to compute an expected value, you need to know all possible outcomes and their respective probabilities. You then multiply each value by its probability and add everything up. Let us do a simple example.

Roll a fair die. The possible outcomes are the numbers 1 through 6, each occurring with probability 1/6, and by what I just described, we get

1 × 1/6 + 2 × 1/6 + 3 × 1/6 + 4 × 1/6 + 5 × 1/6 + 6 × 1/6 = 3.5

as the expected value of a die roll. You may notice that the term “expected” is a bit misleading because you certainly do not expect to get 3.5 when you roll the die. Think instead of the expected value as the expected *average* in a large number of rolls of the die. For example, if you roll the die five times and get the numbers 2, 3, 1, 5, and 3, the average is (2 + 3 + 1 + 5 + 3)/5 = 2.8. If ...

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