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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. 7 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 8 1 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 9 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 11 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? 13 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 14 4 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 15 100 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 16 5