The Lottery Project

So I was curious to figure out how much money the MegaMillions lottery makes. And it occurred to me that you can take a pretty good educated guess. If you want to follow along, download my spreadsheet.

On 22 Apr 2005, the MegaMillions lottery had a winning draw for $205 million USD. The jackpot had accumulated since 04 Mar 2005, climbing an average 24% each draw twice weekly for seven weeks.

The mechanics of the MegaMillions lottery are well-explained elsewhere. A short recap: 52 numbers available for the Mega Number and 52 for the ordinary numbers. There are five ordinary numbers drawn and one Mega Number, for a total of 135 145 920 possible tickets. The MegaMillions pays out not only for the jackpot, should it be hit, but also for various combinations of Mega Number and ordinary number:

• all five numbers (without the Mega Number) pays $175 000,
• four numbers plus the Mega Number pays $5 000,
• four numbers (without the Mega Number) or three numbers plus the Mega Number pays $150,
• two numbers plus the Mega Number pays $10,
• three numbers (without the Mega Number) pays $7,
• one number plus the Mega Number pays $3, and
• the Mega Number alone pays $2.

The jackpot can be paid out either as a lump sum or as an annuity over 25 years (with 26 payments in all). Although the jackpot was billed as $205 million, the actual payout as a lump sum would not be the full amount because of the time value of money. Instead, the jackpot is discounted to its fair present value using a discount factor. I am not sure what this number is, but I guess it is on the order of 4-6%.

So all the facts are available to make an estimate of the costs of this lottery, and these are the costs of the non-jackpot payouts every draw and the jackpot in the last draw. The number of non-jackpot payouts in each category over the whole period is obtainable from the Megamillions site. The value of each ticket of each also listed. With these two sets of facts, we can see that the ordinary payouts came to $71 253 244. And by assuming the discount factor to apply to the jackpot at 6%, we calculate the jackpot cost at $108 676 462.02. So the payouts come to $179 929 706.02.

But what about the revenue? How can you make a guess at how many tickets were sold from which the $180 million was paid out? Well, we know the number of combinations that win for each payout type. If we know how many tickets of a certain variety paid out each draw, then we can estimate the fraction of the universe of tickets that were in play, and therefore the number of tickets sold.

For example, on 22 Apr 2005, there were 25 tickets that had five ordinary numbers and no Mega Number. There are 51 possible tickets in 135 145 920 that can win, so a fair estimate of the tickets sold is:

= 135 145 920 x (25 / 51)
= 135 145 920 x 49.0196%
= 66 248 000

So we can repeat this for each of the draws and estimate the number of tickets sold at 395 111 150. By my reckoning, this means that the sales less cost of goods sold amounts to $215 181 444, which is not too shabby.

Using the spreadsheet, you can play with other assumptions and come up with other numbers. For example, you can change the discount factor for the jackpot. You can use a different way of estimating the number of tickets sold. (It's arguable that the number of tickets with the Mega Number and no ordinary numbers is a pretty good estimator of the number of tickets sold because there is less variability expected in this large number.) Still, I think that you'll find the total payout is not greatly more than 50% of the revenue.

Excel spreadsheet
Access database of results

Durango Bill's Applied Mathematics: Mega Millions Odds
MegaMillions site

© 2001-8 Hugh Brown. All rights reserved.