Math: Expected Value

Tutoring math, you might be asked about statistics, in which expected value is an early topic.

In many everyday situations, the expected value is equal to the mean, aka the average.  The difference is more in definition than in practice.  Expected value involves probability, whereas mean involves fact.

Formally, the expected value is defined as follows:

Expected Value=Σoutcome_value*probability

The Σ symbol means “sum”.  In other words, the expected value takes each possible outcome and multiplies it by its probability.  Then, it adds all those products together.

Example 1: Give the expected value of rolling a fair six-sided die.

Solution:  We know that the probability of getting each result is 1/6.

The expected value, E, of the die roll is

E=(1/6)1+(1/6)2+(1/6)3+(1/6)4+(1/6)5+(1/6)6=(1/6)(1+2+…+6)=21/6=3.5

Notice that the expected value is not necessarily a possible value. Not surprising, really: if you take the average height of ten people, you’ll likely arrive at a height that none of them is.

Example 2: Find your expected payoff in the following situation: 1000 people pay $1 each for a ticket. Then a number is drawn. The holder of the winning ticket gets $500.

Solution: The probability of winning is 1/1000, in which case you get $500. Really though, the payoff is only $499, since you had to spend $1 to buy the ticket. The probability of not winning is 999/1000, in which case you get nothing. Once again, you still had to pay $1 to lose, so the payoff is -$1. Applying our definition of expected value we get

E=(999/1000)(-1) +(1/1000)(499)=-0.999+0.499=-0.50

So, your expected payoff is -$0.50. It had to be a loss, since not all the money paid in was awarded.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Leave a Reply