Math 141: Business Mathematics I
Fall 2015
§8.2 Expected Value
Instructor: Yeong-Chyuan Chung
Mean and Expected Value
We are familiar with finding the average, or mean, of a set of numbers. Your GPA is one
such average.
Example. Records kept by the chief dietitian at the university cafeteria over a 25-week period
shows the following weekly consumption of milk (in gallons):
Gallons of milk 200
Number of weeks
4
201
6
202
8
203
5
204
2
The average number of gallons consumed per week can be computed by:
200 · 4 + 201 · 6 + 202 · 8 + 203 · 5 + 204 · 2
.
25
In this example, we can rewrite the expression as
200 ·
4
6
8
5
2
+ 201 ·
+ 202 ·
+ 203 ·
+ 204 · .
25
25
25
25
25
In this form, we are thinking of the number of gallons of milk as a random variable, and
we are multiplying each possible value of this random variable by its relative frequency or
probability, and then we add these together.
More generally, given a random variable X, if x1 , x2 , . . . , xn are the possible values of X with
associated probabilities p1 , p2 , . . . , pn , then the expected value of X is given by
E(X) = x1 p1 + x2 p2 + · · · + xn pn .
Notice that all the information required to compute E(X) is contained in the probability
distribution associated with the random variable X.
The expected value of a random variable X is a measure of the central tendency of the
probability distribution associated with X. As the number of repetitions of the experiment
gets larger and larger, the average of the observed values of X will get closer and closer to
the expected value of X.
1
2
§8.2 Expected Value
Example (Exercise 8 in the text). If a sample of three batteries is selected from a lot of ten,
of which two are defective, what is the expected number of defective batteries?
Example (Exercise 15 in the text). In a lottery, 5000 tickets are sold for $1 each. One first
prize of $2000, 1 second prize of $500, 3 third prizes of $100, and 10 consolation prizes of
$25 are awarded. What are the expected net earnings of a person who buys one ticket?
Example (Exercise 20 in the text). The proprietor of Midland Construction Company needs
to choose one of two projects. He estimates that the first project will yield a profit of $180,000
with a probability of .7 or a profit of $150,000 with a probability of .3; the second project will
yield a profit of $220,000 with a probability of .6 or a profit of $80,000 with a probability of
.4. Which project should the proprietor choose if he wants to maximize his expected profit?
§8.2 Expected Value
3
Odds
We often speak of the odds in favor of, or the odds against, an event occurring, such as
the odds that a certain team will win the SEC Championship. Let us define these concepts
precisely.
The odds in favor of an event E occurring is given by
The odds against an event E occurring is given by
P (E)
(assuming P (E) 6= 1).
P (E c )
P (E c )
(assuming P (E) 6= 0).
P (E)
Whenever possible, odds are expressed as ratios of whole numbers. Notice that the odds in
favor of E and the odds against E are reciprocals of each other. If the odds in favor of E
a
are , then we say that the odds in favor of E are a to b. If the odds against E occurring
b
b
are , then we say that the odds against E are b to a.
a
Example (Exercise 34 in the text). The probability of an event E occurring is .8. What are
the odds in favor of E occurring? What are the odds against E occurring?
If we are given the probability of an event occurring, then we can compute the odds in favor
of E and the odds against E using the definitions above. Sometimes we are given the odds
instead and we want to compute the probability. If the odds in favor of an event E occurring
are a to b, then the probability of E occurring is
P (E) =
a
.
a+b
Example (Exercise 40 in the text). If a sports forecaster states that the odds of a certain
boxer winning a match are 4 to 3, what is the probability that the boxer will win the match?
§8.2 Expected Value
4
Median and Mode
Besides the mean, there are two other measures of central tendency of a group of numerical
data. These are the median and the mode.
The median of a group of numbers arranged in increasing or decreasing order is
• the middle number if there is an odd number of entries;
• the average/mean of the two middle numbers if there is an even number of entries.
The mode of a group of numbers is the number that occurs most frequently.
A group of numerical data may have no mode, a unique mode, or more than one mode.
Example (Exercise 44 in the text). The frequency distribution of the hourly wage rates
among blue-collar workers in a certain factory is given in the following table.
Wage rate ($)
Frequency
10.70 10.80 10.90 11.00
60
90
75
120
11.10 11.20
60
45
Find the mean wage rate, the mode, and the median wage rate of these workers.
Example (Exercise 49 in the text). The weights, in ounces, of ten packages of potato chips
are as follows: 16.1 16 15.8 16 15.9 16.1 15.9 16 16 16.2
Find the mean, median, and mode of these weights.