Calculating Expected Value

Expected Value, E(X), of a Random Variable X

Start with an Experiment.

List the Outcomes:

O1, O2, ¡­ On

With each outcome is associated a probability:

p1, p2, ¡­, pn

and a value of the random variable, X, :

X1, X2, ¡­, Xn

The expected value of the random variable X is, by definition:

E(X) = p(O1)X1 + p(O2)X2 +p(O3)X3 + ¡­. +p(On)Xn

The expected value is often denoted just by E.

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Calculating Expected Value

Make a table like this one

Outcome Probability Value of X

O1

HH

1/4

7

X1

O2

HT or TH

1/2

3

X2

O3

TT

1/4

-15

X3

-2

? 1?

? 1?

? 1?

E = ? ? 7 + ? ? 3 + ? ? ( ?15 ) =

4

?4?

?2?

?4?

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Fair Games; Expected Value is 0

In a fair game,

E = p(win)*winnings +p(lose)*loss = 0

Example: Suppose for some game, p(win) = 2/6; p(lose) = 4/6

If you lose, you pay $1; if you win other player pays you $D

What should D be if the game is to be fair?

E=

2

4

* D + * ( ?1)

6

6

Set E = 0

D=2

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Expected Value - Example

p(no Ace) = 0.659; p(at least one A) = 0.341

? The game costs $2 to play. You are dealt a poker

hand. If it contains an Ace you get your $2 back, plus

another $1. What is the (expected) value of the game

to you?

Outcomes Probability Payoff to you

At least

one ace

No aces

0.341

$1 + ($2 - $2)

0.659

-$2

E = 0.341*($1) + 0.659(-$2) = -$0.978

value of

game to10

player

2

E as a function of payoffs and cost of game

? Same game as before. costs $2 to play. You are

dealt a poker hand. If it contains an Ace you get your

$2 back, plus another $1.

O utcom es P robability P ayoff

W in

0.341

Lose

0.659

$1 + ($2 - $2)

= $3 - $2

-$2

E = 0.341*($3 - $2) + 0.659(-$2)

Rewriting this, we get

= 0.341*($3) +0.659($0) + (0.659 + 0.341)(-$2)

= 0.341($3) +0.659($0) - $2

= p(win)*winnings +p(loss)*0 - cost of game

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E = expected value of winnings - cost of

game

Outcomes Probability Winnings Cost of

game

Win

0.659

$3

Lose

0.341

$0

$2

12

3

Insurance Example

? An insurance company charges $150 for a policy that

will pay for at most one accident. For a major

accident, the policy pays $5000; for a minor accident,

the policy pays $1000. The $150 premium is not

returned.

? The company estimates that the probability of a

major accident is 0.005, and the probability of a minor

one is 0.08.

? What is the expected value of the policy to the

insurance company?

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Insurance Example - 2

Outcomes

Probability

major

accident

0.005

minor

accident

0.08

no accident

1- 0.005-0.08

= 0.915

Cost to

company

Premium

- $5000

$150

- $1000

$0

E = 0.005(-$5000) + 0.08(-$1000) +0.915($0)+ $150 = $45

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Multiple Choice Tests

(a) (b) (c) (d) (e)

¡°Your grade = # of correct answers - (1/4)(# of incorrect answers)¡±

Or, you get 1 point for each correct answer, and ¨C(1/4)pt

for each incorrect answer.

Suppose you guess at the answer to a question.

What is the expected number of points you¡¯ll get for

that question?

Outcome

Probability

Value

guess right

1/5

1 point

guess wrong

4/5

-(1/4) point

E = (1/5)*1 + (4/5)*(-1/4) = 0

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100 Questions Multiple Choice Test¨C

5 foils, different scoring

¡°Your grade = # of correct answers - (1/5)(# of incorrect

answers)¡±

Suppose you guess at the answer to all 100

questions. What is the expected grade for the test?

Per question:

Outcome

Probability

Value

guess right

1/5

1 point

guess wrong

4/5

-(1/5) point

E = (1/5)*1 + (4/5)*(-1/5) = 0.04

For the test: 100*0.04 = 4

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