Expected value and fair games

What you will be able to do

• find the expected value of a game from its outcomes and their probabilities;
• decide whether a game is fair, and find the prize that makes it fair;
• explain the house edge of a casino game and confirm it by simulation.

A game at a school fair

A made-up stall at a school fair has a spinner with $8$ equal sectors. One play costs $2$ tokens. One sector pays $10$ tokens, two sectors pay $3$ tokens each and the other five pay nothing. Should you play?

Imagine $8$ plays in which every sector comes up once. You pay $8 \times 2 = 16$ tokens. You win $10 + 3 + 3 = 16$ tokens. On average you win back exactly what you pay.

The same sum, written with probabilities, is the expected prize:

The play costs $2$ tokens, so the expected gain is $2 - 2 = 0$ tokens.

Expected value

For an outcome that takes the values $x_1, x_2, \dots$ with probabilities $p_1, p_2, \dots$, the expected value is

It is not the value you expect on one play. You never win exactly $2$ tokens on this spinner. It is the average per play in the long run, by the law of large numbers. The next chapter, on random variables, builds on this idea.

A game is fair when the expected gain is $0$. The fair game above favours neither the player nor the stall.

The house edge

A roulette wheel of the European kind has $37$ pockets: the numbers $1$ to $36$ and a green zero. Eighteen of the numbers are red. A bet of $1$ token on red wins $1$ token if the ball lands on red, and loses the token otherwise. The expected gain is

The green zero is what makes the game unfair. On average a player loses about $2.7$ tokens for every $100$ tokens bet. This is the house edge. On a single evening a player may win. Over millions of bets the casino's share is very steady.

Try it in Python

The program plays the fair spinner and the bet on red $100000$ times each, and prints the average gain of each.

Making a game fair

Suppose the top sector of the spinner paid $W$ tokens instead of $10$. The game is fair when the expected prize equals the cost:

So $\dfrac{W}{8} = 1.25$, and $W = 10$ tokens. A bigger prize favours the player, and a smaller one the stall.

Your turn

The usual mistakes

• Forgetting the cost. The expected prize is not the expected gain. Take away what you pay to play.
• Expecting the expected value on one play. It is a long-run average, and often not a possible result.
• Leaving out the outcomes that pay nothing. Their probabilities still matter, because all the probabilities must add up to $1$.
• Thinking a long run of losses must turn. A casino game has no memory, as the lesson on streaks showed.

Remember these

• $E = x_1 p_1 + x_2 p_2 + \dots$, the long-run average.
• Expected gain = expected prize minus the cost. A fair game has an expected gain of $0$.
• A negative expected gain for the player is the house edge. Over many plays the average settles near it.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A fair die is rolled. You win $6$ tokens for a six and nothing otherwise. What is the expected prize?

2. The game in question 1 costs $1.5$ tokens to play. What is the expected gain? Is the game fair?

3. A made-up game pays $4$ tokens with probability $0.25$ and $1$ token with probability $0.5$, and nothing otherwise. Find the expected prize.

4. On the European wheel, a player bets $1$ token on red $1000$ times. About how much should the player expect to lose?

5. Why can a casino be sure of a profit while each player cannot be sure of a loss?

References

• OpenStax, Introductory Statistics, Chapter 4, the section on mean or expected value.