Lottery Expected Return Formula (How to Calculate It)

Recently I wrote a post about full pay video poker. And it started my brain down the road of the lottery expected return formula.

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One of the things I like about this kind of game is that understanding how to calculate expected return for lottery games is relatively easy. It requires a minimal understand of probability. But once you understand how to calculate the expected return, you also know how to calculate the house edge for lottery games.

I don’t see a lot of pages in Google that explain these concepts clearly, so I’m going to explain them here.

What Is the Lottery Expected Return Formula?

Here’s the lottery expected return formula:

Total expected return = the expected return for each prize.

And, the expected return for each prize is the payout for that prize multiplied by the probability of winning that prize.

The mathematical principle of expectation is one way of looking at a gambling game. The lottery expected return formula enables you to take that look.

The expected return is the long-term outcome for every possible result. It tells you what your long-term payback is in a specific lottery game. The calculation is based on only 2 factors:

1. The probability of each result
2. The payout for each result

When you’ve completed the calculations, you’ll be left with a percentage. This percentage is the expected return for the lottery game.

For example, if you’re playing in a lottery game with a 50% expected return, a $1 ticket is worth 50 cents. You multiply the expected return by the cost of the ticket to get its expected value.

And keep in mind that this is an average over a long period of time. In most lottery games, it’s impossible to lose 50 cents on a $1 lottery ticket. You’ll usually lose the whole dollar. If you win, it’ll be some amount between $1 and $100,000 or more.

When you play tens or hundreds of thousands of tickets and average all the wins and losses together, your actual results will resemble the mathematical expectation most of the time.

How to Calculate Expected Return for Lottery Games

You multiple the probability of each result by the payout for each result. To get the expected return for all the games, you add all those returns together to get the total return for the game.

When you do this, you know what the actual worth of a lottery ticket is mathematically.

Here’s an example probability table calculating the expected return for a specific lottery game:

Lottery Result	Payout	Probability	Percent Probability	Return (Payout X Percent Probability)
Match 5 numbers of 5	$100,000	1 in 575,757	0.000174%	17.37%
Match 4 numbers of 5	$300	1 in 3387	0.029525%	8.86%
Match 3 numbers of 5	$10	1 in 103	0.970874%	9.71%
Match 2 numbers of 5	$1	1 in 10	10.000000%	10.00%
Total Expected Return				45.94%

The game I used for that example is Rolling Cash 5, which is a traditional draw game from the Ohio Lottery. To play, you choose 5 numbers from between 1 and 39, and a ticket costs $1.

How Changing Jackpots Affect the Expected Return

One interesting thing about this calculation is this:

As the size of the top jackpot gets bigger, the expected return increases, too.

When there’s a drawing, and no one wins, the jackpot gets bigger. This increases the expected return for the game accordingly.

In the Rolling Cash 5 example from the table, you can see that 17.37% of the total return for the game comes from the jackpot of $100,000.

If you double the size of that jackpot to $200,000, you double the return for that jackpot to 34.74%.

This also increases the overall expected return for the game.

In fact, if the jackpot gets large enough, the game becomes a positive expectation bet.

How to Calculate the Break Even Point for the Game

In the Rolling 5 example, you might decide you’re only going to play when you have at least a positive expectation.

How do you get from there to here, though?

To calculate the break even point for the game, you start by subtracting the returns for all the other prizes from 100%.

This gives you a 28.57% return for all the prizes that aren’t the jackpot.

This means you need a big enough jackpot to get a 100% return. Subtracting 28.57% from 100% gives you 71.43%.

You also know that every $100,000 of the top prize is the same as 17.37% in return, so you just divide 71.43% by 17.37%.

That’s 4.11.

Multiply 4.11 by $100,000, and you have the breakeven point for the game — $411,000.

If the jackpot is higher than $411,000, you’re playing a positive expectation game.

But this isn’t as great a deal as you might think.

This Is Largely Academic, Though

If you’re thinking you now have the secret to getting rich from the lottery, think again.

Just because you know how to get to a positive expectation bet doesn’t mean that you’ll be a winner.

My friend Michael Bluejay points this out on this page: Expected Value Is Useless When Applied to the Lottery.

He points out that it’s a fallacy to think that a game with a high house edge is a horrible waste of money. It’s not a horrible waste of money because the amount you’re wagering and losing is minimal compared to what you’d gamble and lose at a casino game like blackjack, craps, or roulette.

That’s because you’ll make dozens of bets per hour at any of those casino games. Those will probably be $5 bets or more, too.

With the lottery, most players don’t bet more than $5 per week. They’ll lose less money in a week playing the lottery than the average blackjack player will lose in a few hours at the casino.

More relevant to this discussion is when Bluejay discusses mistake #2 — thinking that waiting to play the lottery when the expectation becomes positive makes a difference.

That’s basically because the odds of winning that top jackpot are so long that you could play for the rest of your life and easily never hit the jackpot.

And a return on a jackpot that you never hit isn’t really a return, is it — regardless of how big it is.

Michael explains all of this better on his site, so be sure to read the page I linked to.

How to Calculate the House Edge for Lottery Games

The house edge for a lottery game is easy to calculate. You just subtract the expected return from 100%.

If the expected return is 45.94%, for example, the house edge is 55.06%.

That’s huge when compared to the house edge for games like blackjack, craps, roulette, slot machines, or video poker.

But those games with the smaller house edge are most costly for the average gambler for two reasons:

1. Casino gamblers are making more bets in a shorter time period.
2. Casino gamblers are betting more per bet.

The expected loss on a gambling is the amount of action multiplied by the house edge.

The amount of action is the number of bets multiplied by the size of those bets.

If you’re a roulette player betting $10 per spin, and you make 35 bets per hour for 2 hours, you’ve put $700 into action.

With a house edge of 2.70% on a single zero roulette game, you’re looking at an expected loss of $18.90.

With a house edge of 55.06%, you’d need to buy $35 worth of lottery tickets to expect to lose that kind of money. I don’t know anyone who buys 35 lottery tickets at a time — although I’m sure they’re out there.

Conclusion

It’s good to understand the lottery expected return formula because you know what you’re dealing with. My dad told me to never gamble on a game I didn’t understand.

And I want to understand a game thoroughly.

With the lottery, as with slots or video poker, you can’t understand the game thoroughly unless you understand how the game becomes profitable.

I hope you step away from reading this post with a good understanding of how to calculate the expected return for lottery games. If I left something out, or if something isn’t clear, please leave a comment and let me know.