EXAMPLE 1 The Tri-State Daily Numbers

Here is a simple lottery wager: the “Straight” from the Pick 3 game of the Tri-State Daily Numbers offered by New Hampshire, Maine, and Vermont. You pay $0.50 and choose a three-digit number. The state chooses a three-digit winning number at random and pays you $250 if your number is chosen. Because there are 1000 three-digit numbers, you have probability 1-in-1000 of winning. Here is the probability model for your winnings:

What are your average winnings? The ordinary average of the two possible outcomes $0 and $250 is $125, but that makes no sense as the average winnings because $250 is much less likely than $0. In the long run, you win $250 once in every 1000 bets and $0 on the remaining 999 of 1000 bets. (Of course, if you play the game regularly, buying one ticket each time you play, after you have bought exactly 1000 Pick 3 tickets, there is no guarantee that you will win exactly once. Probabilities are only long-run proportions.) Your long-run average winnings from a ticket are

or 25 cents. You see that in the long run the state pays out one-half of the money bet and keeps the other half.