Table of Contents |
Expected value is the long-term average taken by the outcomes from a chance experiment. It can sometimes be a confusing term, so we can also use the term mean or average of a probability distribution.
The expected value is found by adding the products of each individual outcome and its probability.
The spinner below has three 1's, a 2, two 3's, and two 4's.
The probability distribution is shown here. The outcomes are 1, 2, 3, and 4, and their probabilities are 3/8, 1/8, 2/8, and 2/8, respectively.
Number | 1 | 2 | 3 | 4 |
---|---|---|---|---|
Probability |
When trying to determine an expected value (what is expected to happen), this expected value will be discussed as a long-term average value that the spinner might produce.
So if the spinner was spun eight times, we would expect three 1's, one 2, two 3's, and two 4's to be obtained. The law of large numbers says that the longer this is done (in the long term), this is exactly what will happen. About three out of every eight times, you'll get a 1. About one out of every eight times, you'll get a 2, etc. Therefore, you're going for the long-term average. Now, if that's the case, the mean is just going to be the average value from those eight spins:
The expected value, also called the mean of a probability distribution, is found by adding the products of each individual outcome and its probability. In this case, 2.375 was the result.
It is not a possibility on the spinner, nor is it an integer. Since the focus is about the long-term average, it doesn't actually need to be an integer. It doesn't need to be possible, either.
Consider the roulette wheel below. There are 38 sectors: 18 red, 18 black, and 2 green. All the sectors are equally likely.
What is the mean of a distribution showing the payouts and probabilities from betting red on a roulette wheel? If you bet $1 and you win, you win $1. If you don't win, you lose the dollar that you bet.
Winnings | -1 | 1 |
---|---|---|
Probability | 20/38 | 18/38 |
There are 18 red sectors on a roulette wheel. There are 20, therefore, that aren't red, out of the 38. Find the expected value for a play on the roulette wheel.
What this means is that over the long-term, you lose about a nickel every time you place a $1 bet on a roulette wheel. Because it's negative, that means you will be losing money over the long-term.
Source: THIS TUTORIAL WAS AUTHORED BY JONATHAN OSTERS FOR SOPHIA LEARNING. PLEASE SEE OUR TERMS OF USE.