# How to Analyze the Lottery

Updated Feb 16, 2006.

## Abstract

The number of winners in a lottery has an approximate Poisson distribution, and the expected number of winners is the number of tickets sold divided by the number of ways to win (approximately 146.1 million for the Powerball Lottery). As the size of the jackpot increases, so does the expected number of winners. As the expected number of winners increases, your expected winnings decrease because you have to split the jackpot with other winners.

Even for very large jackpots, the expected value of a ticket is still less than the price of the ticket.

## How the Lottery Works

The specific lottery we will analyze is the Powerball Lottery run by the Multi-State Lottery Association for about 30 states. The same principles apply to any lottery.

The specific analysis is for the drawing Wednesday, February 15, 2006 because that was when one of us was called by a newspaper reporter and we updated this web page.

The lottery has two kinds of prizes. There are a number of small fixed prizes ranging from \$3 to \$200,000. Then there is the big jackpot, which starts at about \$10 million (we are unable to find the exact number under the current rules) and grows until somebody wins it.

About half of every dollar bet goes to prizes. The other half goes to overhead (administration and advertising) and to the state governments. Of the 50 cents of each dollar that goes to prizes, 19.71 cents goes to pay off fixed prize winners and the rest, the powerball FAQ says, thirty cents of every dollar sold, goes into the jackpot. The more people bet and the more drawings that go by with no winner, the larger the jackpot gets.

We don't have much interest in the fixed prizes. The only thing you need to know about them is that you expect to win 19.71 cents in various fixed prizes per dollar bet (on average, in the long run).

Your chance of winning the jackpot with one ticket is about 1 in 146.1 million (exactly 1 in 146,107,962). If you win, you may have to divide the jackpot with other winners. The jackpot is divided equally among the winning tickets.

To make things more confusing there are two ways you can take your jackpot winnings. You can take your winnings immediately in one lump payment, or you can take your winnings in 30 equal annual payments spread out over 29 years. The first is called the cash option, the second the annuity option.

The advertised value of the annuity option is usually almost twice the size of the cash option because of the interest that builds up over 29 years. The lottery buys US government bonds to cover the annual annuity payments. Thus the difference between the cash and annuity options depends on the interest rate.

How much the annuity is actually worth is a question for an economist, not a statistician. What an economist would call the present value of the annuity is much less than its advertised value, because if you had that much in cash and invested it (say in US government bonds like the lottery does so there is low risk), you would get a lot more back in equal annual payments over 29 years. On the other hand, the present value of the annuity is more than the cash option, because you pay less in taxes on the annuity, and the interest on the annuity is earned before taxes. To keep things simple, we will consider that the present value of the jackpot is the value of the cash option. That understates the value a bit, but is closer than taking the advertised value of the annuity.

## An Overly Simplistic Analysis

As mentioned above, we are going to use the the drawing Wednesday, February 15, as our specific example. The day before (when this was written) the lottery estimated the jackpot would be \$300 million as an annuity or \$145.7 million cash.

If you could be sure that if you won you would be the only winner, the calculation of your expected winnings on a \$1 ticket would be simple. The jackpot is \$145.7 million, and your chance of winning is 1 / 146,107,962. Thus your expected jackpot winnings are

\$145.7 million / 146.1 million = \$0.9972

There is also the possibility of winning one of the fixed prizes. As we said above, your expected winnings for those are about 19.71 cents. Hence your total expected winnings (jackpot plus fixed prizes) are \$1.19 on a \$1 bet. Or at least that's what they would be if you could be sure you wouldn't have to split the jackpot with any other winners. Since you can't be sure of that, this analysis is wrong.

## A Better Analysis

How much you win depends on how many other winners you have to split the prize with. When the jackpot is small and the number of tickets sold is much smaller than 146.1 million, an analysis like the one above wouldn't be far wrong, because the chance of multiple winners is negligible. When the jackpot is huge and the number of tickets sold is a sizable fraction of 146.1 million, multiple winners are likely, and we need a better analysis.

How much you win if you win depends on how many winners there are. So we need to calculate the probability distribution of the number of winners. (More pedantically, we need to calculate the conditional probability distribution of the number of other winners given that you win. Fortunately we don't need to worry about the distinction, because both distributions happen to be the same.)

This distribution is called the Poisson distribution. The formula for the Poisson distribution contains a parameter m which is the expected number of winning tickets, which is the number of tickets sold times the probability of winning (1 in 146.1 million).

Thus the first thing we have to figure out is the number of tickets sold. That's hard. Usually, you can't find this number either in the lottery advertising or in any of the news coverage. However, the reporter who talked to one of us said the lottery expects to sell 80 million tickets this week.

Now we can calculate the expected number of winners, which is

80 million / 146.1 million = 0.548

Now using the formula for your expected jackpot winnings derived on the Poisson distribution page

W = J (1 − em) ⁄ m

where

• W is your expected jackpot winnings,
• J is size of the jackpot,
• m is expected number of winners, and
• e is base of the natural logarithms (use the ex key on a scientific calculator to calculate em).

Plugging in the numbers gives (remember that we are considering the cash value of the jackpot to be its true worth)

\$145.7 million × (1 − e− 0.548) ⁄ 0.548 = \$112.2 million

If this seems a little hard to believe, let's calculate the expectation by brute force not using the formula. The following table gives the relevant numbers.

Number of other winners Probability Amount you win (millions of dollars) Product
0 0.5784 145.7 84.269
1 0.3167 72.85 23.07
2 0.0867 48.57 4.211
3 0.0158 36.42 0.576
4 0.0022 29.14 0.063
5 0.0002 24.28 0.006

The last column is the product of the second and third -- amount you win in each case times the probability of that case. The sum of the last column is the expected value \$112.2 million, which is the same as given by the formula.

The table is also of interest because it gives the probabilities of multiple winners. Recall that we said that the Poisson distribution (the probabilities in the second column) not only gives the conditional probability of the number of other winners besides yourself given you win it also gives the unconditional probability of the number of winners. Thus we see that the most probable number of winners was zero (59% chance), the next most probable number was one (32% chance), and the next was two (9% chance).

Now the rest of the calculation is easy. Your expected jackpot winnings are \$112.2 million if you win, your chance of winning is 1 in 146.1 million, the expected value of your \$1 ticket is

\$112.2 million / 146.1 million + \$0.1971 = \$0.9651

or 96.5 cents on the dollar (19.8 cents from fixed prizes and 76.8 cents from the jackpot).

This is not bad considering that if you take the annuity, the present value may be somewhat underestimated by the cash value, which would make the expected value even higher. It is also not bad considering what you expect for a drawing with a small jackpot. Then the cash option may be as small as \$7.3 million, with very few tickets sold (compared to 146.1 million) we can use our crude estimate of the expectation that ignores the possibility of multiple winners

\$7.3 million / 146.1 million + \$0.1971 = \$0.2471

just 25 cents on the dollar (19.8 cents from fixed prizes and 5.0 cents from the jackpot).

## Summary

Summing up our conclusions, it's much better to play in a week with a big jackpot than a small one (duh!) but even in a week with a near record jackpot (second largest in history) the lottery is only almost but not quite a break even proposition.

For the record, there was no winner (the most likely outcome with 57.8 percent chance), and the jackpot was even bigger for the next drawing.

Questions or comments to: John Corbett corbett@stat.umn.edu or Charles Geyer charlie@stat.umn.edu