Science

Don’t Just Look at the Odds: The Math Says You Should Buy a Mega Millions Ticket

A woman's hand holds out four Mega Millions tickets. On the counter below her hand are a small stack of $20 bills.
Mark Ralston/Getty Images

The Mega Millions jackpot has reached a staggering $1.6 billion, meaning Tuesday’s drawing will be the largest in U.S. history. The odds haven’t changed, with your chances of winning at 1 in 302 million, and so the value of a single $2 ticket has only increased.

As we wrote when the jackpot was a mere $970 million last week, you can do the math by multiplying the size of the winnings (very large) by the probability of you winning (very low) to get the value of your ticket. For this new $1.6 billion jackpot, ignoring tax implications, the value of a ticket comes out to be $5.53—a very good deal.

We wrote last week that the complications of tax brackets and the variability of state taxes meant that you were probably making a mathematically smart bet by buying a ticket. The conclusion was that, to be absolutely certain the expected value of your ticket was worth more than its cost, you’d have to do a bit of research into your own state’s tax and lottery rules. Well, that’s no longer true. There’s no way to slice the numbers where you don’t come out with a good deal.

Here is how we (with the help of some experts) did the math:

The value of a ticket, as laid out by Jordan Ellenberg, can be found by multiplying the probability of getting a certain prize by the value of the prize itself, repeating for all potential prizes, and then adding your results. If we consider just the jackpot, we multiply the incredibly large prize ($1.6 billion) by the incredibly small odds of winning (1 in 302 million) and get an expected value of $5.29—definitely a lot more than the $2 cost of a ticket.

But you’re not just competing for a jackpot when you buy a ticket. There are other prizes for if you get some of the numbers but not all, ranging from $1 million to just $2. When you plug those numbers with their respective probabilities (you can see them here) into the equation, you get an even better value for your ticket: $5.53. That’s a great deal.

And even when you factor in federal taxes at the highest tax bracket of 37 percent, which you’ll have to pay if you hit the jackpot, and state taxes at the highest level, you won’t come close to dipping down to that $2 mark. With the highest federal tax rate and an 8 percent state tax, your ticket is still worth $3.23.

One assumption in this model is that you’ll take the annual payment option and receive the full $1.6 billion. If you were to take the cash option—projected at the time of this writing at $904 million—your expected value could fall below $2. But you’d really only choose that if it made financial sense for you (say, you would invest in it better than the state would), so we shouldn’t think of that as lowering the value of your ticket. (To answer a common question about this post: the other way to lower the value of the ticket below its price is to assume a split pot. It is true that extremely high number of players in this lottery increases the probability of multiple winners, but the odds are still so incredibly low that we don’t believe it should factor into your decision.)

The naysayers will point to the powerful odds against you when telling you not to buy a ticket. And that’s a good reason not to buy more than one or two tickets, given you’re almost certain not to win. But the expected value is clear: Take your chances, savor your mathematically sound risk-taking, and indulge in some daydreaming without feeling like a dolt.

Update, Oct. 23, 2018, at 11:50 a.m.: This post has been updated to address questions about the possibility of multiple winners.