x x x n - University of Houston

Section 7.2
Expected Value
The average (mean) of n numbers, x 1 , x 2 , x 3 ,…, x n is x .

x = x1 + x2 + ... + xn
n

Example 1: Find the average of the numbers: 12, 15, 16, 20, 25.

Example 2: Find the average of the results of this experiment given the frequency table:

x Frequency of x
12 4
15 2
16 1
20 2
25 1

Sometimes, we’ll need to find the average when we aren’t given a frequency table.
Suppose we are just given the probability distribution of a random variable X. In that
case, all we’ll have to work with is a list of values, each with its associated probability.
Using the frequency table shown in Example 1, we can construct a probability
distribution of the random variable.

x P(X = x)
12 0.4
15 0.2
16 0.1
20 0.2
25 0.1

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Expected Value of a Random Variable X
Let X denote a random variable that assumes the values x 1 , x 2 , x 3 ,…, x n with associated
probabilities p1 p 2 , …, p n , respectively. The expected value of X, E(X), is given by

E( X ) = x1 p1 + x2 p2 + ... + xn pn
The expected value of a random variable X is a measure of the central tendency of the
probability distribution associated with X . In repeated trials of an experiment with
random variable X, the average of the observed values of X gets closer and closer to the
expected value of X as the number of trials gets larger and larger. Geometrically, the
expected value of a random variable X has the following simple interpretation:
If a laminate is made of the histogram of a probability distribution associated with a
random variable X , then the expected value of X corresponds to the point on the base of
the laminate at which the latter will balance perfectly when the point is directly over a
Fulcrum.

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Example 3: (Recall from 7.1) Let X denote the sum of the numbers on the upper faces
when two dice are cast. Find the expected value E(X).

x P(X = x)
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36

Example 4: Find the expected value of the random variable X given the following
distribution.

x -5 -2 0 1 4 6
P( X = x) 0.1 0.1 0.2 0.2 0.2 0.2

Example 5: A box contains 10 quarters, 7 dimes, 4 nickels, and 2 pennies. A coin is
drawn at random from the box. What is the mean of the value of the draw?

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Example 6: An investor is interested in purchasing an apartment building containing six
apartments. The current owner provides the following probability distribution indicating
the probability that the given number of apartments will be rented during a given month.
Number of
Rented Apt 0 1 2 3 4 5 6
Probability 0.02 0.08 0.10 0.12 0.15 0.25 0.28
a. Find the number of apartments the investor could expect to be rented during a given
month.
b. If the monthly rent for each apartment is $799, how much could the investor expect to
collect in rent for the whole building during a given month?

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Odds in Favor of and Odds Against
If P(E) is the probability of an event E occurring, then

1. The odds in favor of E occurring are P(E) , P(Ec) ≠ 0 .
P( E c )

2. The odds against E occurring are P(Ec ) , P(E) ≠ 0.
P(E)

Note: Odds are expressed as ratios of whole numbers.

Example: If P(E) = 2 , then P(Ec ) = 3 and P(E) = 2/5 = 2.
5 5 P(Ec ) 3/5 3

The odds in favor of E occurring are 2 to 3.

The odds against E occurring are 3 to 2.

Example 7: The probability that it will rain tomorrow is 0.2. What are the odds that it
will rain tomorrow? What are the odds that it will not rain tomorrow?

Probability of an Event (Given the Odds)

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 8: The odds that it will rain tomorrow are 1 to 4. What is the probability that it
will rain tomorrow?

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Example 9: The odds in favor of a certain baseball team wining the pennant are 2:7.
What is the probability that that team will not win the pennant?

Example 10: The odds against Laura winning a certain raffle are 99:1. What is the
probability that Laura will win the raffle?

Example 11: The probability that the race horse Black will win a race is 0.86. What are
the odds in favor of Black winning?

Example 12: Sam believes that the odds of him getting a job are 5 to 2. What is the
(subjective) probability that he gets a job?

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