world. The bad part should be clear. Each win ($10) does not make up for each loss ($11). In the long run,
we would expect to lose money. We would not want to play this game. In this same example, if we only
pay $10.01 when we lose, the game is not nearly as unbalanced. It has a higher expectation.
Example 3.12:
In this game, when we win, we are paid $10. When we lose, we pay $10. Would we want
to play this game?

8
If you have already answered (Yes or No), you have answered too soon. You
don’t know what the
probability of winning is. If your theoretical chance of winning is 50%, this is a fair game and playing is
fine. If your theoretical chance of winning is 49%, the game is unfair, but not too unfair. If your
theoretical chance of winning is 40%, the game is very unfair and playing would be very bad.
Based on these examples, we see that your theoretical expectation is a combination of your probability
of winning along with what you get paid when you win and what you pay when you lose.
Definition 3.6:
The
mean
of a discrete random variable
X
, is defined by the formula
x
Support
xf x
(provided the sum converges)
( )
.
.
.