How to spot a positive expected value bet in the lottery
The June 8, 2024 Lotto Powerball Case Study
You generally should expect to lose money when you gamble. If commercial operators are offering you a bet, it’s because over time, they expect to take your money.
But sometimes a bet can be in your favour.
Games are often governed by rules that, under certain circumstances, make betting advantageous. When these specific conditions are met, the bet can yield a positive expected value, meaning a higher likelihood of winning money.
In New Zealand, that happened on 8 June 2024 with the Lotto Powerball lottery. For every $1 you wagered, you expected to win back $1.30.
Must-be-won roll-downs guarantee large payouts
Betting the Powerball doesn’t usually make much sense given the very low odds of you winning it. Sure, the payout may be $40m, but your chances of winning don’t justify the purchase. If we assume the Powerball jackpot is $40m, in New Zealand’s Lotto Powerball you purchase lines of numbers for $1.50 per line, for a 1 in 38,383,800 chance of winning, or an expected value of ($0.46).
That is, you expect to get $1.04 back for betting your $1.50. That gets bumped up a bit by lower division prizes, but it’s still overall a negative bet.
But once the Powerball jackpot hits $50m, it must be won.
If no one wins division 1 Powerball, it rolls down to division 2 and is split among those winners. Division 2 has a 1 in 6,397,300 chance of being won per $1.50 line. If no one wins Division 2, it goes to Division 3 and so on…
The key insight is that if the prize is must be won, it’s actually pretty straightforward to work out if a bet is positive expected value. Estimate the cost of the total number of tickets sold, and compare that to the total payout. If the total payout is greater than the cost of the total number of tickets sold, then it is a positive expected value bet.
To see this, imagine everyone bought their tickets and then in the hour before the draw, they formed a syndicate. The syndicate agrees to split all prize money in proportion to how much they had spent on tickets. Everyone would get more money than they put in.
So let’s estimate the number and total cost of tickets sold.
Estimating the cost of the total number of tickets sold
From previous must-be-won draws I had a reasonable estimate of how many tickets are sold. Powerball Lotto does not report the total number of tickets sold, but they do report the winners of each division. For example, division 7 has 1 in 352 odds of being won. It had 91,419 winners.
Lotto report the odds of winning any division with a single line of numbers.
We can convert those odds to percentages, then simply add them to calculate the total percentage chance that a single line wins any prize. That comes out to 0.334%, or a 1 in 299 chance of winning any division prize.
Lotto also report (make sure to select 8 June 2024) how many prizes there were for each division.
There were 107,078 total Powerball winners.
We can then estimate the total number of lines purchased by taking the total winners and multiplying it by 299.
That implies that 32,053,477 (299 * 107,078) $1.50 Powerball lines were bought, for a total of $48,080,216 spent on Powerball tickets.
You might have seen the bottom right of the image above, saying that the total Powerball prize pool was $51,929,165. That might make you think there was only an ~8% return from purchasing a ticket. But that number ignores those who didn’t win powerball, but did win a regular lotto prize.
Estimating total payout for all those who purchased Powerball tickets
When you buy a Powerball ticket you are really buying two things. You are buying a regular lotto ticket, plus an additional number (the Powerball) that gives you a chance to win Powerball division prizes if you win a regular division, plus match the Powerball.
The total Powerball prize pool is just showing the winnings to the Powerball aspect of the ticket. So even not counting Lotto winnings, buying a Powerball ticket gave you an 8% expected return (i.e. receive $1.08 for betting $1).
The lottery does not report the Lotto-only prizes won by those who bought Powerball tickets. They just report the Lotto prizes by division:
But we can also use some basic probability to calculate the odds of winning Lotto but not winning the equivalent Powerball division.
For any individual division that is:
which is just:
Summing all divisions, it is:
If we then take the probability of winning each division, Lotto only, then multiply that by the number of Powerball lines bought, we get an expected number of winners for each Lotto-only division:
So now all we have to do is work out the total amount of winnings paid to these estimated Lotto-only winners with Powerball tickets. Using Lotto’s stated payouts for these divisions and calculating total:
That’s another $9,970,509 in expected winnings, bringing the total amount paid out to those who bought Powerball tickets $62,855,676.
When you compare that to the $48,080,215 spent purchasing tickets, the $14,775,461 extra money represents a 31% return.
Ok, but how much should I bet?
Finding positive expected value bets is one thing. But you really shouldn’t bet big on these huge variance outcomes, because you’ll likely go broke before you ever win.
The mathematical way to approach this is by using the Kelly Criterion. The kelly fraction of your bankroll is the amount that maximises the geometric growth rate of your bankroll. If you bet less, you could have grown your bankroll more quickly by taking more risk. If you bet more, you could have grown your bankroll more quickly by taking less risk.
The Kelly formula for a bet with two outcomes is very straightforward. But Powerball has 14 divisions (including non-Powerball divisions). So working out the optimal fraction of your bankroll is far more complicated.
With the help of Claude 3.5 Sonnet 3.5 (new fav AI), I created a simple Python programme to calculate the Kelly fraction of my bankroll.
It told me zero. Hmm, I wondered if something had gone wrong and checked the code carefully, putting in some intermediate steps to present me with the expected value.
Happily, it came up with an identical expected valuation calculation to me, but it was still telling me I should buy zero tickets given my bankroll (I used an estimate of my net worth).
Then I altered the code to calculate what minimum bankroll I would have to have for the optimal Kelly bet to be a single $1.50 Powerball line…
$71,940,917.94
Whoops. Ah well, I’m still going to buy tickets when it’s positive expected value - I refuse to wait until I have $72m to play the lottery.
Lol how many tickets did you end up buying? And how much did the Kelly Criterion suggest you needed to have for that to be an optimal bet?