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Expected Value, E(X), of a Random Variable X
Start with an Experiment. List the Outcomes : O 1 , O 2 , … On With each outcome is associated a probability: p 1 , p 2 , …, p (^) n and a value of the random variable, X, : X 1 , X 2 , …, X (^) n
The expected value of the random variable X is, by definition: E(X) = p(O 1 )X 1 + p(O 2 )X 2 +p(O 3 )X 3 + …. +p(O (^) n )X (^) n
The expected value is often denoted just by E.
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Calculating Expected Value
E = + + =
Make a table like this one
Outcome Probability Value of X HH 1/4 7 HT or TH 1/2 3
TT 1/4 -
O 1
O 2
O 3
X 1
X 2
X 3
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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/ 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 D
Set 0
E
D
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Expected Value - Example
- The game costs $2 to play. You are dealt a poker hand. If it contains an Ace you get your $2 back, plus another $1. What is the (expected) value of the game to you?
p(no Ace) = 0.659; p(at least one A) = 0.
Outcomes Probability Payoff to you
At least one ace
No aces 0.659 -$
E = 0.341*($1) + 0.659(-$2) = -$0.978 (^) value of game to player
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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 Cost to Premium company major accident
minor accident
no accident
E = 0.005(-$5000) + 0.08(-$1000) +0.915($0)+ $150 = $
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Multiple Choice Tests
(a) (b) (c) (d) (e)
“Your grade = # of correct answers - (1/4)(# of incorrect answers)”
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 guess wrong
1 point
-(1/4) point
E = (1/5)1 + (4/5)(-1/4) = 0
Or, you get 1 point for each correct answer, and –(1/4)pt for each incorrect answer.
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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
guess wrong
1 point
-(1/5) point
E = (1/5)1 + (4/5)(-1/5) = 0.
For the test: 100*0.04 = 4
