Transcription of Calculating Expected Value - Mathematics
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Calculating Expected Value - Mathematics
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?=+ 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.
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) = 0.341 Outcomes Probability Payoff to you At least one ace 0.341 $1 + ($2 - $2) No aces 0.659 -$2
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