Nega Millions

The Mega Millions lottery has been in the news a bunch lately, due to the growing jackpot, and I wondered how big the jackpot would need to be for it to be worthwhile to play. Specifically, I was curious about the relationship between the jackpot and the number of players, and exactly how unlikely winning is.

In many articles I found, like the one above, I saw the probability quoted as 1 in about 300 million, but I wanted to go through it myself: As of 2017, the drawing involves 5 numbers chosen from a set of 70, and 1 chosen from a set of 25. The order doesn't matter, so we can write this as

where the exclamation mark is the factorial operator, defined as n! = n*(n-1)*(n-2)*...*2*1. If you do this calculation, you'll find the number given elsewhere – Initially, I thought the order did matter, in which case the 5! would be left out, and there would be significantly more possibilities.

The total jackpot starts at $20 million, and then grows based on the number of tickets sold. We can look at how the jackpot has changed over the past year:

Each time the jackpot is won, it drops back to the baseline $20 million. The slope of the climb depends on the number of players – Note that as the jackpot increases, the slope of the curve also increases. When discussing outcomes with different probabilities, mathematicians often use the expectation value, a weighted average using the probabilities as weights. Since multiple winners split the jackpot, we can write the expectation value for the payout as

The chance of winning is 1/C; the chance of not winning then is (1 - 1/C). If k people out of N players win, we can raise the respective probabilities to the power of k and N-k to compound them. Now we can look at the relationship between the jackpot, number of players, and expected payout:

A couple interesting features of this plot: $500 million appears to be a threshold for a lot of people – The slope of the points increases here, suggesting people outside the set of regular players have joined in. Due to the low chances of winning, the expected value only barely passes $4, although that does exceed the purchase price of $2, so buying a ticket is "worth it" (if you ignore the time cost of buying the ticket and checking the numbers). The low win probability also means that the multiple winner aspect is pretty insignificant. For the numbers I used here, it didn't seem to change anything from the base winning number.

Th other thing I was interested in was how many drawings passed between wins. We can split the data up based on when the jackpot resets to $20 million, and then plot the curves on top of each other:

There are several that last only a couple days, but I was really surprised that the longer-lasting ones have almost identical changes in the number of players over time. That suggests the threshold behavior I mentioned earlier also applies on a smaller scale: As the jackpot grows, there's a continuous increase in the number of people willing to take the chance.

Going into this, I was already confident that playing the lottery was a terrible idea, but to me the expected value really drove that idea home: Even a record-setting jackpot will only barely push you over the break-even point.