Earlier this year, we considered the maths behind deciding when to buy a Powerball ticket.

Now, the Mega Millions jackpot has reached a ludicrously high $US636 million, and so we decided to repeat the analysis for this lottery.

The most basic thing to consider in any game of chance, whether you are looking at a lottery, a roulette wheel, or a deck of cards in a blackjack game, is your expected winnings from the game. This is basically a measure of how much you will win on average by playing the game a large number of times.

The expected winnings for a game can be found if we know the probabilities of the different outcomes in the game, and the earnings (or losses) associated with each outcome. We multiply the probabilities by the earnings or losses, and then add these up.

The expected value will tell us what will happen on average if we play a game a large number of times. If the expected value is positive, then on average we are winning money, and so if we play the game long enough, we should end up in the black. If the expected value is negative, then on average we are losing money, and so we should not be playing this game.

We can find the expected value for playing Mega Millions with the current ridiculous jackpot. In Mega Millions, you pick five different numbers between 1 and 75, and one number between 1 and 15. If all six match, you win the jackpot. If you match some of the first five numbers and/or the last number, you win a smaller prize.

The Mega Millions website helpfully includes the odds of winning either the jackpot, or of winning various smaller prizes. Since the cost of a Mega Millions ticket is $US1.00, we can subtract this from each of the prizes to get our actual take home winnings for each of these outcomes, and so find the expected value of a Mega Millions ticket:

The first column shows each outcome — how many of the five numbers from 1 to 75 we got right, and whether or not we got the last number from 1 to 15. The second column shows the prize and the third the prize less our investment of a dollar per ticket. In the fourth column, we have the odds of getting each outcome, as per the Mega Millions website. In the fifth column, we convert those odds into a probability between zero and one by taking the reciprocal of the odds. Finally, we multiply together our winnings less investment by the probability, and sum these up, giving us the expected value.

Notice that by far our most likely outcome is that we get none of the numbers right. There is about a 93% chance that we just wind up losing our dollar. Despite this, since the jackpot is so enormously high right now, our average winnings are a nice positive $US1.63, indicating that we should consider buying a ticket.

Another notable property of Mega Millions is that the lower prizes do not help us too much. Without that jackpot, the expected value is a very unhappy — $US0.82. So, Mega Millions really is all about the jackpot.

One factor to consider is that, as with most lotteries, there are two options for the prize: the advertised prize of $US636 million is based on a 30 year annuity, receiving the prize in smaller annual chunks for 30 years. The other option is to take cash up front, but at a huge discount. Today’s up front cash prize is $US341 million — a big drop, but nothing to sneeze at. We can find the expected winnings for taking the cash up front:

The expected value of a Mega Millions ticket here is lower — just $US0.49 as opposed to the $US1.63 taking the annuity — but it is still positive, and this is still a viable option.

There are other confounding factors — you also have to consider the effect of taxes, and how that plays with both options. Even though this lottery has a positive expected value, it has an extremely high variance and standard deviation, owing to the fact that there is one very, very unlikely outcome where you do incredibly well, and the overwhelming majority of the time, you just lose a buck.

The most interesting confounding factor is the possibility of multiple winners. How many people are playing will affect the odds of a split and the size of the resulting jackpot (whether it is broken in to two parts, or three, or four, or however many winners there are). We can, however, estimate the expected value of a ticket based on how many tickets have been sold, and thus on the odds of a split, and the consequent value of the jackpot:

The horizontal axis shows the number tickets sold, in millions, and the vertical axis shows the expected value in dollars of a ticket, based on the likelihood of a split, and the size of a split jackpot. The blue curve shows the outcome for taking the annuity prize, and the red line shows the outcome for taking the cash prize.

Here, we can see that the expected value of a ticket stays positive (meaning you should consider buying a ticket) as long as fewer than 730 or so million tickets have been sold. The cash prize fares far worse — you run into expected losses with only 265 million tickets sold.

So, as long as there are fewer than 730 million tickets sold, a fairly likely situation right now, the expected value of a ticket should be positive, and so you should consider buying a Mega Millions ticket today.

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