James P Houghton

# Playing the Lottery

30 Jan 2013

A good rule of thumb for deciding if it makes economic sense to play the lottery is: 'no'. In order for the operators of the lottery to make money, lotteries have to take in more money on average than they pay out:

Number of tickets * Price of ticket > Payout Amount

The statistical return on a ticket, its expected value, is the probability of winning a prize multiplied by the size of the prize itself:

Expected Value = Probability of Winning * Value of Prize

For instance, if I have a 1 in 10 chance of winning the prize, and the prize is 50 dollars, then I could expect, if I played enough games, to win about 5 dollars per ticket:

Expected Value = .1 wins/ticket * \$50/win = \$5/ticket

But for the lottery organizers to make a profit, the price of a ticket must exceed its average expected payout value. In this case they need needs to charge more than \$5 per ticket, and for me to have a statistical return on my investment, I need to spend less than \$5 per ticket. Its a zero sum game.

You'll notice that I keep saying that the lottery needs to bring in more money than it pays out on average. From time to time, a string of drawings will occur with no one claiming the jackpot. In this case, the prize increases to be significantly higher than average, some people think that this brings the expected value above the price of a ticket. Lets take a look:

For the Powerball lottery, there are 9 prizes (excluding power play), one of which wins the jackpot, and the remainder of which have lesser value prizes. Assuming that only 1 ticket wins the jackpot, we get a break-even expected value when the cash value of the jackpot exceeds \$464 Million, or about \$301 Million after tax.
The cash value of comparable jackpot has historically been about 72.2% of the advertised multi-payment jackpot, meaning that statistically, the advertised jackpot must be at least \$642 million before it makes sense to play Powerball.

The MegaMillions Lottery has a similar prize structure:
The historic cash jackpot for the MegaMillions lottery has been 58.7% of the annuity option, meaning that you break even (ignoring other players) if the advertised jackpot exceeds \$392 million. In the last year, a ticket for either of the two lotteries has met this mark only once:

Now, there is a catch: if multiple people have the winning numbers, then the jackpot is split between them, and of course, the higher the jackpot, the more people play. We need to update our chart to include the odds that you win the jackpot and that no one else does, the odds that you win and one other person does, that two other people do, etc.

To calculate the odds that W other people win the lottery out of N other players, we multiply first the odds of the first W players picking the jackpot number with the odds of the next (N-W) players picking a non-jackpot number, and then multiply by all the different orders in which they could have bought their tickets:

Probability of W other winners = (Probability of Winning)^W * (1-Probability of Winning)^(N-W) * (N choose W)

The following chart shows the probability of multiple people winning the Powerball lottery as the number of players increases.
We can also calculate the 'Expected Value' of the number of Powerball winners for a given number of tickets sold:
Nothing too Earth shattering.

To get the expected value of the jackpot with N other players, we multiply your odds of winning by the odds that W other folks win, and by the jackpot you get if W people win, and sum for all possible values of W:

Expected Value = SUM(Probability of Winning * Probability of W other winners * Jackpot/W, W=0..N)

In practice, we only have to sum over the first 10 or so values of W to get a very good approximation. We can now plot the point at which the expected value of a lottery ticket breaks even as a function of the total jackpot and the number of players.
In this case, tickets for any drawing happening to the right of the blue curve would have a statistically positive return on investment. Here is the same calculation for the MegaMillions lottery:
In this chart we see that even the \$640M drawing that broke even when we ignored other players is now a significant distance from the break-even curve, because so many tickets are purchased.

In a future post, we'll look at how you would maximize your expected value, even if you didn't plan on breaking even, by choosing lotteries to participate in carefully. Then, if all works out, we'll see if there are ways to improve that calculation by factoring in other data.

Other resources:
http://www.durangobill.com/PowerballOdds.html
http://www.lottoreport.com/PBSalesbystate.htm
 http://www.lottoreport.com/mm03sales.htm