## AN AVERAGE VALUE FOR A PROBABILITY DISTRIBUTION

To discuss the idea of an average value of a probability distribution, let's discuss Roulette, one of the most popular casino games. In this game, there is a wheel with 38 numbered metal pockets (1 to 36 plus 0 and 00). The wheel is spun moving a metal ball and the ball comes to rest in one of the 38 pockets. The wheel is balanced so that the ball is equally likely to fall in any one of the 38 possible numbers. You play this game by betting on various outcomes --- you win if the particular outcome is spun.

Suppose you decide to bet \$10 on the numbers 1-12. You will win \$20 (and keep your \$10 bet) if the ball lands falls in slots 1-12; otherwise, you lose your \$10. Is this a good game for you? How will you do, on the average, if you play this bet (\$10 on 1-12) many times?

First, let's find the probability distribution for the amount of money you win on a single bet. There are two possibilities -- either you win \$20 or you win -\$10 (win a negative number means you lose +\$10). You win if the ball falls in slots 1-12. Since there are 38 slots, each of which is equally likely, the probability of each slot is 1/38, and so the probability of falling in 1-12 (and you win) is 12/38. The probability you lose is 1 - 12/38 = 26/38. In summary, the probability distribution for the amount you win is

AMOUNT YOU WIN PROBABILITY
20 12/38
-10 26/38

To summarize this probability distribution, we compute an average value, which is often called the mean . We compute this average in two steps:

• we multiply each outcome value by the corresponding probability to get products

• we add all of the products to get the average

We illustrate this computation for the roulette winnings in the table below. In the PRODUCT column we multiply each winning by its probability. The value at the bottom of the PRODUCT column is the average.

AMOUNT YOU WIN PROBABILITY PRODUCT
20 12/38 (20)(12/38) = 240/38
-10 26/38 (-10)(26/38) = -260/38
SUM -20/38 = -.53

Here we compute the average value to be \$-.53 or 53 cents. What does this mean?

• First, it is important to note that the average winning is negative. This is not a fair game and it is to your disadvantage (and the casino's advantage) for you to play this game.

• One useful interpretation of the average winning \$-.53 is that it is approximately the average winning per game if you were to play this game (bet \$10 on numbers 1-12) a large number of times.

Let's illustrate this using a computer to simulate many games. I play the game 100 times one day and I record the winnings (in dollars) for the games in the table below.

```
20   -10   -10   -10    20   -10   -10    20   -10   -10   -10   -10
20    20    20    20    20   -10    20   -10    20   -10   -10   -10
20    20   -10   -10    20   -10   -10   -10   -10   -10   -10    20
20   -10   -10   -10   -10   -10    20   -10   -10    20    20    20
-10   -10    20   -10   -10   -10    20   -10   -10   -10   -10    20
-10    20   -10   -10    20   -10   -10   -10   -10   -10   -10   -10
-10   -10    20   -10   -10    20   -10   -10   -10   -10   -10    20
-10   -10    20   -10   -10    20   -10    20   -10   -10   -10    20
-10   -10    20   -10
```

How have I done at the end of the day? It turns out that I am \$40 in the hole after these 100 games. In other words, I have lost an average amount of \$40/100 = \$.40 or 40 cents on each game. This is close to the average value .53 that I computed from the probability distribution above. I would observe an average loss closer to .53 if I played this game a much larger number of times.

Page Author: Jim Albert (© 1996)
albert@bayes.bgsu.edu
Document: http://www-math.bgsu.edu/~albert/m115/probability/average_value.html