Part of the complete guide: Lottery odds: the complete guide
The idea
Suppose a game pays $10 half the time and nothing otherwise. Its expected value is 0.5 x $10 = $5. If you play many times, you would average $5 per play, even though a single play pays 10 or nothing. Expected value is that long-run average.
A pool lottery makes it simple
In a pool lottery all prizes come from ticket sales, so the maths is neat. If the payout is 90% of sales, then on average each ticket returns 90% of its price. The number of tickets does not change that.
| Item | Value |
|---|---|
| Ticket price | $5.00 |
| Share paid as prizes | 90% |
| Expected prize per ticket | $4.50 |
| Expected net result per ticket | -$0.50 |
| Per $100 spent | about -$10 |
Seeing it with a real draw
Take 1,000 tickets. Sales are $5,000, the pool is $4,500, and all of it is paid out. If you owned all 1,000 tickets you would get $4,500 back for $5,000 and lose $500. Owning one ticket is a one-thousandth share of that, so the expected loss is 50 cents, but the actual result is either a loss of $5 (most of the time) or a gain of $85 to $2,245 (rarely).
Why every lottery has a negative expected value
Someone must run the lottery, and in most lotteries a share of sales goes to costs, taxes, good causes or profit. That share is what makes the expected value negative. A lottery with a 100% payout would have zero expected value and no money for anything else. Where a lottery pays out around half, as many large ones do, the expected loss is closer to half of what you spend.
Variance: why real results differ so much
Expected value is an average, and the spread around it is large. With a 1.4% chance of any prize, you will usually lose your $5, sometimes win a little and very rarely win a lot. Over a handful of plays, results say almost nothing about expected value. That is why a lucky streak does not mean the game is good.
How to use this
- Treat the expected loss as the price of entertainment, like a cinema ticket.
- Compare lotteries by payout share, not by jackpot size.
- Never rely on a lottery to solve a money problem.
- Decide a budget that you would be happy to lose entirely.
Expected value of a $50 weekly habit
| Period | Spend at $50 a week | Expected return at 90% | Expected loss |
|---|---|---|---|
| 1 month (4 weeks) | $200 | $180 | $20 |
| 1 year (52 weeks) | $2,600 | $2,340 | $260 |
| 5 years | $13,000 | $11,700 | $1,300 |
These are averages. In any single year the real result will likely be far from the average, in either direction.
What expected value cannot tell you
- How you will feel about a loss or a win.
- Whether a game is fun for you. That is personal.
- Whether you can afford the loss. Only your budget answers that.
Example: expected value of a $5 pool ticket
Take a draw with 1,000 tickets: pool $4,500, one $2,250 prize, three $450 prizes and ten $90 prizes. A ticket's chances: 1/1000 of $2,250 = $2.25; 3/1000 of $450 = $1.35; 10/1000 of $90 = $0.90. Add them: $4.50. Subtract the $5 price: an expected loss of $0.50. With 10,000 tickets, every prize is ten times bigger and the chances ten times smaller, so the expected value is still $4.50. This is why the number of tickets sold changes the shape of the game, not its average cost.
Using expected value to compare games
| Game | Payout share | Average loss per $100 spent |
|---|---|---|
| Typical large jackpot game | about 50% | about $50 |
| Typical scratch-off | about 65% to 70% | about $30 to $35 |
| Pool lottery paying 90% | 90% | $10 |
| Casino game with 2% house edge, per bet | 98% per bet | $2 per $100 wagered, but wagered many times |
Illustrative figures. Check each game's published payout rate.
Ready?
A weekly draw you can check yourself.
$5 tickets, a public random value, and every result published with the data to recompute it.
Frequently asked questions
Can expected value ever be positive?
Only in unusual situations, such as a very large rollover jackpot in a few games, and even then taxes and shared prizes usually remove the edge.
Does buying more tickets improve expected value?
No. It scales your expected loss in proportion.
What does 90% payout mean?
Of all ticket sales, 90% goes into the prize pool and 10% goes elsewhere.
Is a bigger jackpot better value?
Not by itself. Look at the payout share and your realistic chance.
Can expected value be zero?
Only in a game that returns every dollar it takes in, which leaves nothing for costs.
Why does a bigger jackpot not fix the expected value?
Cash option, tax and prize sharing remove most of the extra.
Is expected value the same as profit?
No. It is an average over many plays, not what you will get.