Expected ValueDate: 12/09/97 at 20:06:08 From: Kerry Nesser Subject: Expected value What is expected value and how do you work it? Date: 12/10/97 at 13:57:11 From: Doctor Tom Subject: Re: Expected value Hi Kerry, Although the concept of expected value is quite general, let me explain it in terms of gambling, where the concept first arose. It means your average expected payoff in the long run. For example, if we play a game where we flip a fair coin, and if it's heads, you pay me a dollar, but if it's tails, I pay you two dollars (a good game for you, right?), then on average, out of every two times we play, you will win once and I will win once. But when you win, you get $2 and when I win, you get -$1 (in other words, you lose a dollar). So in the long run, for every two times we play, you will gain $2 + (-$1) = $1. On average, for every two games, you will win $1. So on average, you will win half that, or 50 cents, every time you play. This is useful, because now it's easy to estimate how much you will win if we play 1000 games. At 50 cents per game, on average, you "expect" to win about 1000 times 50 cents, or $500 over the course of our 1000 game match. To calculate expected value, just add up the sum of the probabilities of each event times the payoff for each event. In the example above, the probability of heads or tails is 1/2 each, so the expected value is: E = 1/2($2) + 1/2(-$1) = 1 - 1/2 = 1/2, or 1/2 dollar. To show how this works in another situation, suppose that again we agree to play a game where we roll a single die, and here's what happens for each result: 1: you win $3 2: nobody wins or loses 3: I win $5 4: you win $3 5: I win $4 6: you win $2 Is this a good game for you? To find out, work out the expected value (to you). Remember that when you win, the payoff is positive, and when I win, your payoff is negative, since you pay me. If the die is fair, every one of the possibilities above has an equal probability of 1/6: E = 1/6(3) + 1/6(0) - 1/6(5) + 1/6(3) - 1/6(4) + 1/6(2) E = 3/6 - 5/6 + 3/6 - 4/6 + 2/6 = -1/6 So on average, you lose 1/6 dollar every time we play, so it's a bad game for you. Every 600 times we play, you will lose, on average, 600(1/6) = 100 dollars. Your expected loss on each game is 1/6 dollar. I hope this helps. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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