The Mathematics of Money

So, if your company manufactured 82,000 of the new phones, and you decided to use a

3.25% expected relative frequency, the expected frequency of defective phones would be:

Expected frequency 
 (3.25%)(82,000) 
 2,665

Expected value is similar to expected frequency, but not exactly the same. The expected

value of a quantity is the average amount you expect that quantity to represent on a per item

basis. This expected value can be calculated by multiplying:

Expected value 
 (Expected relative frequency)(Amount per occurrence)

Suppose, for example, that you are estimating from past experience that the average cost to

your company of a warranty claim for a mobile phone is $127.40. The expected value (per

phone) of warranty costs would then be:

Expected value 
 (3.25%)($127.40) 
 $4.14

The term “expected value” is not always used. It is usual to replace the word value with

whatever the value is supposed to represent. In this example, you might refer to this

expected value as the “expected warranty cost per phone” or some similar description.

Example 16.2.5 A minor league baseball events promoter is offering a promotion

at an upcoming game. One fan will be randomly chosen from the stands and given the

opportunity to bat against the team’s ace pitcher. If the fan hits a home run before

accumulating three strikes, he will win $50,000.

From past experience of similar events, the promoter believes that such a fan will win

the prize one out of every 1,250 times the promotion is run. Calculate (a) the expected

relative frequency of the fan winning and (b) the expected prize for this promotion.

(a) The expected relative frequency would be 1/1,250 or 0.08%.

(b) The expected prize would be

______^1

1,250

($50,000)
$40

The word expected can be misleading. When we say that the expected warranty cost per

phone is $4.14, we certainly don’t mean that we actually expect each phone to generate

$4.14 in warranty costs. Likewise, we certainly don’t mean that we actually expect the

baseball promotion to cost $40 in prize money. In fact, it is very unlikely that an individual

phone would generate $4.14 in warranty costs, and it is impossible that a fan could receive

$40 in prize money from the baseball contest.

The word expected is used here in a technical way. Loosely speaking, it is an average.

The expected cost per phone means that we expect that the average warranty claims

cost per phone will be $4.14, not that any individual phone will generate that cost.

The expected prize means that we expect that, if we run similar contests repeatedly,

the average amount of winnings per contest will work out to be $40, not that $40 will

be paid out in any individual contest. We do not take these values as applying to any

individual occurrence, any more than we would say that since the average American

family has 2.3 children we therefore expect that Bob and Sally Smith will have exactly

2.3 children.

Expected values are useful when interpreted as predicted averages. Knowing that the

expected cost for warranty repairs is $4.14 would tell you that you need to build $4.14 into

the price of each phone to generate the money that you will need overall to cover your

warranty repair and replacement costs. Knowing that the expected prize of the baseball

contest is $40 would help an insurance company to know how much to charge the baseball

promoter for a policy that would pay the $50,000 prize if it is won.^4

(^4) Yes, this is actually common practice. Usually if a promotion like this is offered, rather than take the risk of having to
Copyright © 2008, The McGraw-Hill Companies, Inc.pay a large prize, the contest sponsor actually will purchase a type of insurance to cover the event.
16.2 Measures of Average 623