This has nothing to do with D&D. I was fantasizing about winning the lottery and, as an intellectual exercise, decided to calculate the odds of the various winning tickets. Most people probably think this is very straightforward, but it is actually a bit more involved than it seems. After verifying that my calculations matched those on the lottery website, I gave some thought to how to increase the chance of winning.

The Rules

The official rules are located at the lottery website. Here is a quick overview.

In the Powerball lottery, 59 white balls are placed in a drum. Five are then drawn (without replacement). These are known as the Pick-5 numbers. This gives a total of 600,766,320 possible outcomes. However, lottery tickets do not need to match the balls in the same order that they are drawn. The five numbers drawn can be arranged in 120 different ways 1-2-3-4-5, 1-3-5-2-4, 2-5-1-4-3, etc. Therefore, there are 5,006,386 (600,766,320/120) unique sets of numbers. In addition, 35 red balls are placed in a second drum. One ball, known as the power ball, is drawn from this drum. That gives us 175,223,510 unique lottery numbers. In other words, with a single ticket, you have a one in 175 million chance of winning the jackpot.

A lottery ticket costs two dollars. For an additional dollar, you can add a “Power Play” that increases the value of any winning ticket (I will be ignoring the Power Play option hereafter). Winning combinations and amounts are listed in the following table:

CombinationPayoutPower Play
Three matches\$3x2
Four matches\$100x2
Five matches\$1,000,000x2
PB + Zero matches\$4x3
PB + One match\$4x3
PB + Two matches\$7x2
PB + Three matches\$100x2
PB + Four matches\$10,000x4
PB + Five matchesJackpotx1

The jackpot varies significantly, increasing each time it goes unclaimed. If multiple people have winning tickets for the jackpot in the same drawing, they split the amount.

How To Win The Lottery

The only way to ensure winning is to buy a ticket with every possible combination of numbers. As shown above, there are 175,223,510 unique lottery numbers. At \$2 per ticket, you will need \$350,447,020 to buy them all. In addition to the jackpot ticket, you will also have 5,502,139 other winning tickets worth a total of \$63,166,120 (not counting the jackpot). These are listed below.

# of TicketsCombinationPayoutTotal
486,540Three matches\$3\$3,405,780
9,180Four matches\$100\$918,000
34Five matches\$1,000,000\$34,000,000
3,162,510PB + Zero matches\$4\$12,650,040
1,581,255PB + One match\$4\$6,325,020
248,040PB + Two matches\$7\$1,736,280
14,310PB + Three matches\$100\$1,431,000
270PB + Four matches\$10,000\$2,700,000
5,502,139Tickets Worth\$63,166,120

Note that since it will cost you \$350,447,020 to buy all those tickets, the jackpot needs to pay out at least \$287,280,900 (350,447,020 – 63,166,120) for you to break even. Also, if anyone else matches all five numbers and the Powerball, the jackpot will be split.

Play All The PB Numbers

If you don’t match the Powerball, you will need to match at least three numbers in order to win anything. However, if you do match the Powerball you will win something even if you don’t match anything else. Because of this some people recommend playing all the Powerball numbers. That is to say you should pick five numbers and buy 35 lottery tickets, each with those five numbers but each with a different Powerball number. That absolutely guarantees that one of those tickets will match the Powerball and you will win something. Your odds of winning the jackpot (on whichever ticket matches the Powerball) also drops from one in 175 million to one in 5 million (a factor of 35). The other 34 tickets are guaranteed not to win the jackpot (since they don’t match the Powerball) but can still win something if they match at least three of the match-5 numbers.

This method is essentially the same as buying all the tickets but doing so over multiple drawings (5,502,139 drawings to be exact). The investment each drawing is significantly less (\$70 opposed to \$350 million) and the impact of splitting the jackpot with other winners is less devastating.

Maximizing Matches

I read a book some time back about numbers theory that described how a mathematician determined a method for selecting numbers in such a way that he could pick a few hundred sets and maximize the number of three and four number matches which allowed him to “beat” the lottery. This of course was a very different lottery system with far fewer permutations. Sadly, I don’t remember the name of the book and in any case it didn’t list specific details, nor would it apply directly to this case. It might be interesting to try to replicate his method though, but it would be quite time intensive and any significant return-to-investment ratio is dubious.

-Joshua

Although I love to fantasize about winning the lottery, and occasionally do buy tickets, the odds of winning are far worse than any other form of gambling. From a numbers standpoint, the lottery system provides a statistical wonderland. But any method of attempting to systematically beat the lottery is doomed to failure.