Expected value

At the fairground a ticket costs 2 €. Every twentieth ticket pays back 30 €, otherwise nothing. Is that worth it? A single ticket says nothing about it, because you either win or you do not. It only gets interesting across many tickets, and there is one number for that view: the expected value.

A weighted average

In an ordinary average every value counts the same. In an expected value every value counts as much as it is likely. So you multiply every possible outcome by its probability and add everything up. That is exactly why it is also called a weighted average: rare outcomes pull little, frequent ones pull a lot.

Back to the ticket. With probability $\frac{1}{20}$ you receive 30 €, with $\frac{19}{20}$ you receive 0 €. The expected payout is $30 \cdot \frac{1}{20} + 0 \cdot \frac{19}{20} = 1.50$ €. But you pay 2 €, so on average you lose 50 cents per ticket. Over 100 tickets that is 50 €, and that is exactly what the stall lives on.

Fair games

A game is called fair when the expected value of the profit is exactly 0. Then stake and payout balance out in the long run and nobody has an advantage. For our ticket the expected profit is $1.50 - 2 = -0.50$ €, so the game is not fair. To be fair the ticket could cost at most 1.50 €, or the prize would have to be 40 €.

What this is really for

Expected values are the basis of every insurance policy. If a loss of 5000 € occurs with a probability of 0.2%, the expected loss is $5000 \cdot 0.002 = 10$ €. That is exactly how the premium is calculated, plus administration and profit. For the individual policyholder that is a losing deal on paper, and still sensible: you trade a small certain minus for protection against a rare but ruinous loss.