What an average return per ticket means
Expected value is the average amount a ticket returns if you imagine playing the same game an enormous number of times. It is not what any single ticket will pay. Most tickets pay nothing, and a few pay a lot, and the expected value blends them into one figure.
To compute it you need two things for every prize tier: the prize and the probability of winning it. Multiply each prize by its probability, add the results, and compare the total with the price of the ticket. The general idea is the same one explained in expected value for any decision under uncertainty.
A made-up game, tier by tier
Real lotteries publish their own prize tables, and those change by game and over time, so this article uses an invented game with round numbers. Suppose:
- A ticket costs 1 unit of currency.
- There are 1,000,000 equally likely tickets in a draw, and you hold one.
- The prizes are fixed amounts, with these counts across the million tickets:
| Prize | Winning tickets | Probability of winning that tier |
|---|---|---|
| 500,000 | 1 | 1 in 1,000,000 |
| 1,000 | 10 | 1 in 100,000 |
| 50 | 100 | 1 in 10,000 |
| 2 | 5,000 | 1 in 200 |
Now multiply each prize by its probability. The contribution of each tier to the average is:
- Top prize: 500,000 × (1/1,000,000) = 0.5
- Second tier: 1,000 × (10/1,000,000) = 0.01
- Third tier: 50 × (100/1,000,000) = 0.005
- Fourth tier: 2 × (5,000/1,000,000) = 0.01
The sum is 0.5 + 0.01 + 0.005 + 0.01 = 0.525. A check using totals gives the same answer: the game pays out 500,000 + 10,000 + 5,000 + 10,000 = 525,000 across 1,000,000 tickets, which is 0.525 per ticket.
So the expected return is 0.525 units for a ticket that cost 1 unit. The expected net result is 0.525 − 1 = −0.475, a loss of about 47.5 percent of the stake on average. In this invented game, about 99.49 percent of tickets win nothing at all, because 1 + 10 + 100 + 5,000 = 5,111 tickets win something, and 5,111 ÷ 1,000,000 is about 0.51 percent.
Notice where the average comes from
More than 95 percent of the expected value in this example, 0.5 out of 0.525, comes from the single top prize that almost no ticket wins. That shape is typical of lottery games: the average is held up by a prize that you are very unlikely ever to see.
This has two consequences. First, small wins do very little to the average. Second, the figure is very sensitive to the top prize. Changing that one number changes the expected value dramatically, while the experience of most players, who win nothing or a small prize, barely changes.
Expected value is an average, not a forecast
An expected return of 0.525 does not mean you will get back about half your money. In this game you will most likely get back nothing, and once in a very long while a large amount. The average only describes what would happen across a huge number of plays, a point made under the law of large numbers. Because the top prize is so rare, the average can take a very long time to show itself, and the results of a small number of tickets can be far from it.
The spread of possible results matters too. A game with the same expected value could pay lots of small prizes or one large one, and the experience is quite different. This idea is called volatility and return when comparing games.
Real lotteries are more complicated
Applying the method to a real game takes extra care, because several details change the answer:
- Shared prizes. Many top prizes are shared if more than one ticket wins, which lowers the average value of the top tier. Whether that happens depends on how many people play and how they choose numbers.
- Fixed pools. Some prizes are shares of a pool rather than fixed amounts, so their value changes with sales.
- Tax and payment terms. Depending on where you live, a prize may be taxed, or a jackpot may be offered as a lump sum that differs from the advertised headline figure.
- Rollovers. When a jackpot carries over, the headline amount rises, but the other items on this list still apply.
Because of these details, a published jackpot figure alone does not give you the expected value. If you want to calculate one for a real game, use the operator’s official prize table and rules, and treat the result as an estimate.
What the average tells you
In games designed to raise money, the operator keeps a share of sales for costs and good causes, so the average return on a ticket is typically below its price. A game where players lose on average is described as having negative expectation, and the arithmetic does not change if you pick different numbers or buy at different times. It also means that spending more on tickets increases the expected loss in proportion.
Because the numbers describe averages, they cannot tell you whether the next ticket will win. Past draws do not affect future ones. If you decide to play, think of the ticket price as the cost of entertainment, keep it within a limit you set beforehand, and do not chase losses with more tickets. If gambling is causing you difficulty, a local support service can help.