Page 94 - Probability Demystified

CHAPTER 5 Odds and Expectation 83

5 5
4. Let A ¼ 5 and B ¼ 31; then PðE Þ¼ ¼ :
5 þ 31 36
3 3
5. Let B ¼ 10 and A ¼ 3 then PðE Þ¼ ¼ :
3 þ 10 13

Expectation

When a person plays a slot machine, sometimes the person wins and other
times—most often—the person loses. The question is, ‘‘How much will the
person win or lose in the long run?’’ In other words, what is the person’s
expected gain or loss? Although an individual’s exact gain or exact loss
cannot be computed, the overall gain or loss of all people playing the slot
machine can be computed using the concept of mathematical expectation.
Expectation or expected value is a long run average. The expected value is
also called the mean, and it is used in games of chance, insurance, and in
other areas such as decision theory. The outcomes must be numerical in
nature. The expected value of the outcome of a probability experiment can be
found by multiplying each outcome by its corresponding probability and
adding the results.
Formally defined, the expected value for the outcomes of a probability
experiment is EðXÞ¼ X PðX Þþ X PðX Þþ þ X PðX Þ where the X
1
1
2
n
2
n
corresponds to an outcome and the P(X) to the corresponding probability of
the outcome.
EXAMPLE: Find the expected value of the number of spots when a die is
rolled.
SOLUTION:

There are 6 outcomes when a die is rolled. They are 1, 2, 3, 4, 5, and 6, and
each outcome has a probability of 1 of occurring, so the expected value
6
1
1
1
1
1
1
of the numbered spots is EðXÞ¼ 1 þ 2 þ 3 þ 4 þ 5 þ 6 ¼
6
6
6
6
6
6
21 ¼ 3 1 or 3:5:
6 2
The expected value is 3.5.
Now what does this mean? When a die is rolled, it is not possible to get 3.5
spots, but if a die is rolled say 100 times and the average of the spots is
computed, that average should be close to 3.5 if the die is fair. In other words,
3.5 is the theoretical or long run average. For example, if you rolled a die
and were given $1 for each spot on each roll, sometimes you would win $1,