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

O2 HT or TH 1/2

O3 TT

1/4

7 X1 3 X2 -15 X3

E

=

1 4

7

+

1 2

3

+

1 4

(

-15)

=

-2 4

8

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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 * D + 4 * (-1)

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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 0.659

$1 + ($2 - $2) -$2

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

value of

game to10 player

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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.

Outcomes Probability Payoff

W in Lose

0.341 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

$2

Lose

0.341

$0

12

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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

major accident minor accident no accident

Probability 0.005

Cost to company

- $5000

0.08

1- 0.005-0.08 = 0.915

- $1000 $0

Premium $150

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 ?(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? 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 guess right 1/5 guess wrong 4/5

Value 1 point -(1/5) point

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

For the test: 100*0.04 = 4

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