Calculating Expected Value - Michigan State University

1 17 Expected Value , E(X), of a Random Variable XStart with an the Outcomes: O1, O2, .. OnWith each outcome is associated a probability:p1, p2, .., pnand a Value of therandom variable, X, :X1, X2, .., XnThe Expected Value of the random variable Xis, by definition:E(X) = p(O1)X1+ p(O2)X2+p(O3)X3+ .. +p(On)XnThe Expected Value is often denoted just by Expected Value ()-73 151112E = + + = 4244 Make a table like this oneOutcome Probability Value of XHH 1/4 7HT or TH 1/2 3TT 1/4 -15 O1O2O3X1X2X329 Fair Games; Expected Value is 0In a fair game,E = p(win)*winnings +p(lose)*loss = 0 Example:Suppose for some game, p(win) = 2/6; p(lose) = 4/6If you lose, you pay $1; if you win other player pays you $DWhat should D be if the game is to be fair?

2 =+ 24**(1)66ED==Set 02ED10 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) = ; p(at least one A) = Probability Payoff to you At least one ace $1 + ($2 - $2)No aces -$2 E = *($1) + (-$2) = -$ of game to player311E 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 $ Probability $1 + ($2 - $2)= $3 - $ $2E = *($3 -$2) + (-$2) Rewriting this, we get= *($3) + ($0) + ( + )(-$2)= ($3) + ($0) - $2 = p(win)*winnings +p(loss)*0 - cost of game12E = Expected Value of winnings - cost of gameOutcomes ProbabilityWinningsCost $3$ $0413 Insurance Example An insurance company charges $150 for a policy that will pay for at most one accident.

3 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 , and the probability of a minor one is What is the Expected Value of the policy to the insurance company?14 Insurance Example - 2 Outcomes Probability Cost to Premium companymajor accidentminor accidentno $5000$150- $1000E = (-$5000) + (-$1000) + ($0)+$150= $451- $0515 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.

4 What is the Expected number of points you ll get for that question?Outcome Probability Valueguess rightguess wrong1/54/51 point-(1/4) pointE = (1/5)*1 + (4/5)*(-1/4) = 0Or, you get 1 point for each correct answer, and (1/4)pt for each incorrect 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 Valueguess rightguess wrong1/54/51 point-(1/5) pointE = (1/5)*1 + (4/5)*(-1/5) = the test: 100* = 4