For those who buy into the lottery, purchasing multiple tickets—one every week, say, or five-or ten-at-a-time—might appear to substantially increase their odds. But what are the odds of winning a lottery? And do any “strategies” really work? To answer that, Book of Odds delves into the numbers behind the biggest American lottery: the Powerball.
The odds a state participates in the Powerball Lottery are 1 in 1.67, or 60% (31 states participate as of the end of October, 2009). Here’s how Powerball works.
Twice a week, the Multi-State Lottery Association holds a drawing in Orlando, Florida. Six numbers are randomly drawn, five from a drum of 59 white balls, and then a sixth from a drum of 39 red balls. To be a grand-prize winner, a ticket must match these six numbers. The order of the five white-ball numbers does not matter, but the red-ball number must match. What are the odds of winning the jackpot? Using common statistics, it’s easy to fall into a couple of traps when trying to figure it out. Two of the following calculations are wrong for one reason or another, and give incorrect odds. One calculation is right on the money.
a) 1/(59x59x59x59x59x39) = 1 in 27,882,047,661
b) 1/(59x58x57x56x55x39) = 1 in 23,429,886,480
c) 1/[(59x58x57x56x55/5!)x39] = 1 in 195,249,054
It’s not A. Those are the odds of winning a lottery in which every number-ball drawn is thrown back into the drum and possibly drawn again. A lottery drawing of that sort could have repeated numbers, like “32 4 17 56 17 17 (38).” In lotteries like Powerball, every number drawn means one less possibility for the next number drawn. (The red ball has a constant 39 possibilities, though, being in a separate drum.) The number of possibilities for white balls drops by one as each ball is drawn. Hence the streak “59x58x57x56x55,” which is then multiplied by 39 (the red ball’s number of possibilities). Which must mean B is correct, right?
Wrong. Because in Powerball, or almost any lottery, for that matter, the order of the numbers doesn’t matter. If the winning numbers are “33 21 47 2 58 (9),” then a winning ticket could read “21 58 2 47 33 (9)” or “47 33 2 58 21 (9).” The number of possible white-ball combinations is 5!, or 5x4x3x2x1. So to adjust for the white-ball order’s not mattering, the equation is divided by 5! So the odds of winning a Powerball jackpot are C, 1 in 195,200,000 (These odds are rounded from above). Better than the first two, but still incredibly small. To put it in perspective, the odds are better that a randomly chosen American male happens to be George Clooney.
Plus, the odds are almost 88% that in a given Powerball drawing, no one will win. Using a statistical function called a Poisson distribution, one can calculate the odds that a drawing will have 0 winning tickets, or 1 winner, or 2, or however many you like. Using the posted Powerball average of 13.5 winning tickets per year—meaning, out of 104 drawings a year, an average 0.1298 per drawing—one can determine that the odds of a drawing having 0 winners are 88%. Above that, the odds of there being:
- 1 winning ticket = 11%
- 2 winning tickets = 0.73%
- 3 winning tickets = 0.03%
So why do people buy lottery tickets? Well, for one thing, a jackpot’s size operates like a feedback loop: it gets bigger, so more people buy tickets, so it gets bigger, so more people buy tickets, so it gets bigger...until someone wins and lives happily ever after, or doesn’t. For another, many people believe purchasing tickets often, or in bunches, substantially increases their chances of eventually winning the lottery. It doesn’t. Even if purchasing 195,249,054 tickets over a lifetime guaranteed a jackpot win (which it doesn’t), that lifetime would have to be very long, or the buyer very rich to begin with:
- Buying two tickets a week, one for every Powerball drawing? Make sure you live 1,877,366 years.
- If that’s out of the question, simply buy 47,000 tickets every week for 80 years.
So live 200 times longer than the world’s oldest tree, or blow $47,000 a week, and you’ll eventually win. Right?
Wrong again. Buying a large amount of tickets doesn’t ensure anything. The only way to guarantee a win is to buy every possible group of winning numbers, all 195 million+, for a given drawing. Which, in Powerball is impossible for a number of reasons. A “brute force” approach like this has been done before, but in a different lottery, under different circumstances. In 1992, Stefan Klincewicz led a 28-person team in purchasing 80% of the number combinations for an Irish Lottery drawing. They went on to win around £1 million Irish, which (after expenses) meant a marginal profit. This strategy was only possible because the Irish Lottery was at the time a 6/36 drawing: 6 balls out of 36 total, in one drum. The odds of winning were 1 in 1,947,792. Since tickets cost £0.50 each, one could—and Klincewicz did—buy about £1 million in tickets and guarantee multiple wins (since the “syndicate” also cashed in on a number of smaller prizes). The Irish lottery has since increased the number of lotto balls to prevent this sort of thing from recurring. In Powerball, however, it just wouldn’t work. Having the money to purchase $195 million worth of tickets overrides the need to win a lottery at all, not to mention the suspicion one might arouse purchasing, say, 50,000 tickets at a gas station.
So what can you do to improve your chances in Powerball? Some websites claim that avoiding the numbers 31 and under—those human-chosen numbers corresponding to birthdays, or lucky numbers (7 & 13)—ups your odds, but this doesn’t improve your chances of winning. It simply improves your chances, ever so slightly, of not sharing the jackpot if you win. This is not probability so much as game theory. And buying more than one ticket for a single drawing will mathematically improve your odds, but they will still be so long that the effect will really be to drain your wallet.
Lottery wins are ultra-rare, and often disastrous—many winners lose friends and family, or go overboard and quickly end up in debt. And by investment-to-return ratio, lottery tickets are one of the worst investments on the planet.
(Note: In future articles we will explore the relative value of different lotteries and the benefits people derive beyond expected financial value.)
For those interested in the mathematics:
The “!” indicates a factorial, which means “multiply by all natural numbers below this one.” So 5!—or “five factorial”—gives 5x4x3x2x1. Factorials are good for calculating the number of combinations a list of objects/events can go in. If you want to watch 4 movies, for example, the number of different combinations in which you can watch them = 4!
Probability of (X wins) = (nˆX)(eˆ−n)/X!, where n = 0.1298 average winning tickets per drawing, X = possible number of winning tickets per drawing, and e = about 2.71828 (the base of natural logarithms).
P(0) = 0.12980(0.8783)/0! = 0.8783 = 88%
P(1) = 0.12981(0.8783)/1! = 0.1140 = 11%
P(2) = 0.12982(0.8783)/2! = 0.0073 = 0.73%
P(3) = 0.12983(0.8783)/3! = 0.0003 = 0.03%