Download 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
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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.
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
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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
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 –(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
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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