Expected Value

Date: 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/