Let me remove all suspense right at the top: The answer is no—I am not buying any Powerball tickets.
As a mathematician, I am often asked whether it’s worth it to play the lottery. How big does the jackpot need to get before I should play? The kind of money at stake in this week’s Powerball drawing—an estimated jackpot of $1.5 billion—is staggering. The folks behind Powerball wanted this to happen: They changed the rules of the game last fall (for the eighth time!) in response to a phenomenon known as “jackpot fatigue.” And it worked, at least for now. There are fewer winners and more huge jackpots, and the hype over this week’s record jackpot has driven ticket sales (and hence profits, from Powerball’s point of view) through the roof.
And yet it’s hard to understand how much a billion is. It sure sounds big, though. Here’s a thought experiment, just to try out your intuition for these things. Don’t calculate, just guesstimate. How long does a million seconds take? How about a billion seconds?
Meanwhile, you’re still wondering if you should play. Answer No. 1 is no, of course you shouldn’t, because you are not going to win. Answer No. 2 is yes, you should, if you want to and it’s fun for you and you have enough disposable income to spend on this particular entertainment.
Since I am on a roll of stating the obvious, let me point out that nobody needs a mathematician to tell them these things. So why do people keep asking me?
I think it is because they are playing anyway, and they want me to tell them it’s OK. They want a rationalization; they want to think that the jackpot is so big that it’s “worth it” to play. Not only is it fun, it’s not foolish! Nobody wants to be foolish after all. To this, I say, see Answer No. 2.
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I do not mean to suggest that there is no interesting mathematics involved in the lottery. There is. So let’s talk math for a few paragraphs.
It is reasonable to think that the jackpot might be “big enough” to make playing worthwhile. After all, the bigger the jackpot, the more your ticket is potentially worth, and so the more you can consider your ticket to be worth on average. The average value of a thing is something we all have an intuitive understanding of, and it’s something we learned how to calculate in grade school. You simply average all the possibilities, weighted according to their probability of occurrence.
For instance, imagine playing a carnival game where you pay some money to roll a die. If you roll a six, you win $6. Otherwise you win nothing. There are two possible payouts, $0 and $6, but that doesn’t mean the average value is $3, because the possibilities are not equally likely. The average value in fact is ($6)(1/6)+($0)(5/6), because the $6 payout occurs with probability 1/6 and the $0 payout occurs with probability 5/6.
This comes out to $1. If it costs you less than a dollar to play the game, then I’d call it a good bet. That doesn’t mean you’ll win. In fact, you’ll lose five times out of six. But it does mean that if you played many times, you’d be very likely to come out ahead. The rare victories would more than compensate for the frequent losses. And if the game costs more than a dollar to play? It’s not a good bet.
We can try to compute the average value—also known as the expected value—of a lottery ticket in a similar way, but it gets a bit complicated. Neither the payouts nor their probabilities are obvious. For instance, in a Powerball drawing, there are smaller prizes ranging from $4 to $1,000,000 in addition to the big jackpot. If you do win the big jackpot (whose value is known approximately, although it changes based on the number of tickets sold), there is the possibility that you’ll share it with someone who played the same lucky numbers. So one possible payout is half the jackpot, and how are you supposed to figure out the probability of that happening?
Once you know the parameters of the game (i.e., for Powerball there will be five balls chosen out of 69, and then one “Powerball” chosen out of 26), there are essentially two factors that affect the average value of a ticket. As the jackpot gets larger, of course, the average value also grows. On the other hand, as more tickets are sold, the average value shrinks, because the chance of jackpot-sharing goes up. And of course these two variables are not independent: The reason the jackpot goes up is that more tickets have been sold.
One way to try to estimate the average value of a ticket is to create a mathematical model of how many tickets are sold as a function of the advertised jackpot. If we understand how these two variables are connected, we can use the advertised jackpot to estimate ticket sales and then use both bits of information to calculate the average value.
I actually did an analysis like this once, in a 2010 study with Skip Garibaldi. We found that very large games, like Powerball and Mega Millions, will never have an average value greater than the $2 ticket price. In a nutshell, this is because no matter how big the jackpot gets, sales patterns indicate that people will buy so many tickets that the average value is pushed down, due to the likelihood of jackpot-sharing. Indeed, seven of the top 10 lottery jackpots in U.S. history—all of which have come since 2007—were shared by at least two winners.
We were surprised in the same 2010 study to discover that there are some small state lotteries that occasionally do offer good bets. For instance, Lotto Texas once had a drawing in which the average value of a $1 ticket was $1.30! This happened because the jackpot had rolled over several times, and yet there weren't enough people playing to significantly increase the chance of jackpot-sharing.
Does that mean you should wait around for those “favorable” drawings and then play them when they happen? Buying a ticket in that Texas lottery would be like playing the carnival game I described above, if it cost about 77 cents to roll the die. The average payout is $1, so it’s a good deal, right? But remember, if you only play once, you’re still likely to lose! You only come out ahead if you are able to play over and over again. Same goes for these lotteries: We found that you’d have to play that “advantageous” Lotto Texas game about 200,000 times before you’d reap the benefits of the good bets.
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People continue to play the lottery in record numbers, despite the common knowledge that it’s not a mathematically sound decision. So this leaves us to ponder a tough and interesting question: Why?
In one sense, the answer is easy. Most of our decisions are not purely rational. They are also emotional, and this particular choice is no different. People play because it’s fun.
But I think there’s more math here, mixed in with the psychology. One of the reasons we are so mesmerized by giant lottery jackpots is that most of us don’t really comprehend what the numbers actually mean, in real terms. Consider this: If you have one shot at it, is it more likely that you can guess my social security number or the winning Powerball numbers? Don’t calculate—just use your intuition.
How about this one: What if I told you that back in 2008, I set an alarm to go off at some point during the Obama administration. The alarm was programmed to last for one second. Are you more likely to guess the exact correct time that the alarm is set for or the winning Powerball numbers? You have to get the right year, month, day, hour, minute, and second, or else you lose.
Or this one: I’ll give you a fair coin, and you flip it 30 times. Is it more likely that you’ll get tails every time or that you will correctly guess the winning Powerball numbers?
All of these things are easy enough to calculate. It’s even more fun to consider things that can’t be calculated exactly: Are there more different Powerball tickets or blades of grass on a football field? Stars in the Milky Way? Powerball tickets or human cells in your body? Specks of dust in your house?
The point of all this is that for most of us, our intuition doesn’t tell us very much, even when numbers differ by many orders of magnitude. In our everyday experience we very rarely encounter numbers of these magnitudes. The numbers are just too big to truly comprehend.
Unfortunately, this can be a serious problem. Politicians can make the cost of the Iraq war, or Obamacare, or the Troubled Asset Relief Program seem large one day and small the next, to suit their political purpose. Do you remember which of those cost the most or the least? Were the price tags off by orders of magnitude, or were they relatively close? It’s hard for most of us to keep these numbers in perspective.
By the way, a million seconds is about 11-and-a-half days. A billion seconds is almost 32 years.
There are about 292 million different possible lottery tickets. That’s 2.9 x 108. There are one billion (109) different nine-digit numbers, although not all of them are legal social security numbers—this site calculates that there are currently 745,395,443 possible SSNs. The number of seconds in eight years is about 252 million. The odds of flipping 30 straight tails from a fair coin are slightly worse than one in a billion. So, all of these numbers are reasonably close to each other—within a factor of 4 or 5.
The numbers that require estimation are, of course, more difficult. Estimates vary, but the number of blades of grass on a football field is probably comparable to the number of Powerball tickets. There are a few hundred billion stars in the Milky Way, so that’s 1,000 times larger. Your body has trillions of cells, so yet another thousand times larger (give or take). I don’t know how dusty your house is. To learn how to make estimates like these, check out the book Guesstimation by Lawrence Weinstein and John A. Adam. They figure it would take 15 40-ton trucks to haul all 292 million Powerball tickets!
So, should you play the lottery? If you’re looking for a mathematical answer from a mathematician and your question is, “Is the jackpot big enough to make the lottery a good bet?” then the answer, I’m afraid, is no.
There is no sense in which you “should” play, but that doesn’t mean buying a lottery ticket is a bad idea. If you don’t have a lot of disposable income, then yes, it is absolutely a bad idea to play! But if you do have some cash you’re looking to spend, and you think it’s fun to play, then go for it. Just keep in mind that you won’t win.