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So you want to win the lottery?

It's easy enough, but just how likely is it?

The likelihood of an event occurring is called its probability. If you are familiar with probability, skip on to the next paragraph (although anyone familiar with probability already knows not to waste their money on the lottery). Betting on the toss of a coin is easily calculated: an honest coin has two different faces, each of which is equally likely to land face-up. There are two possible outcomes, either you win (heads) or you lose (tails). The odds are one to one. Consider tossing two coins and betting on double-heads. There are four possible outcomes (head-head, head-tail, tail-head, tail-tail). You have one chance of winning and three chances of losing. The odds are expressed as "one in four". Tossing an honest die has six equally likely outcomes, so there is one way you can win and five ways you can lose. The odds of winning are "one in six".

In most current lotteries, you must correctly choose six non-recurring numbers between 1 and 49. To calculate the probability of winning, we need to divide the number of possible permutations (how many different six-number sequences are there) by the number of combinations (how many different orders of drawing those six numbers):

    (49 * 48 * 47 * 46 * 45 * 44) / (6 * 5 * 4 * 3 * 2 * 1) = 13,983,816

You have roughly one chance in fourteen million of winning a particular single drawing of the lottery. Fourteen million is just a number, we need to find out what it actually means: is it "good" or "bad"? Let's consider some other probabilities we all face.

Some 150 people died in North America in 1997 due to food poisoning caused specifically by rusty can openers. The population then was around 272 million, so the odds on dying from food poisoning caused by rusty can openers are 272,000,000 / 150 = 1,813,333 to one, roughly one in two million. How does that compare to winning the lottery? Divide 1,813,333 into fourteen million and you get 7.7. Dying because of a rusty can opener is nearly eight times more likely than winning the lottery.

Here's another one. There were 7,792 traffic fatalities in Germany in 1998. The population was then around 82 million, so my chances of dying in a traffic accident were 82,000,000 / 7,792 = 10,523 to one. Roughly one chance in ten-and-a-half thousand of being killed during a year. Divide that into fourteen million and you get 1,329. Dying in a traffic accident is thirteen hundred times more likely than winning the lottery.

Say you buy your lottery ticket on Monday morning for the Saturday evening draw. It stays in your pocket for five and a half days. The chance of your dying in a traffic accident on that day is one-three hundred sixty-fifth of ten-and-a-half thousand, or 3,840,895. Five and a half times that daily chance is one in 698,344. Divide that into fourteen million, and you get 20.02. Being killed in traffic with a not-yet-drawn lottery ticket in your pocket, is twenty times more likely than that the ticket is a winner.

Time for something more cheerful. How about roulette? Say you go into a casino and put down your lottery stake on a random number. Your chance of winning is 1 in 37 (I play European roulette which has only one Zero) or 0.027. Now say you do this and win four times in a row: 1/37 * 1/37 * 1/37 * 1/37 works out to 1,874,161 to one. Roughly one chance in two million. Divide that into fourteen million, and you get 7.46. Hitting the right number four times in a row at roulette, is more than seven times more likely than winning the lottery. (American roulette, with two Zeroes, is significantly worse: four in a row there is only six times more likely than winning the lottery.)

Consider also how much you win. Many people forget (or choose to ignore) that the lottery payout is divided equally among all those who picked the right numbers. There was an admittedly extreme case here in Germany four years ago, when the lottery drew the same six numbers that a TV "tipster" had recommended. Over two thousand people had followed his advice. The 4,100,000 Deutschmark jackpot ("Wow! Four million!") was worth less than two thousand Marks each. Put the same 10 Deutschmark lottery stake on a roulette number, and it returns 360 Marks. Leave it there, the second win returns 129,600. That you must split: in most casinos you may not bet more than 25,000. You put 104,600 aside and leave 25 thousand on your lucky number. The third win returns 900,000 Marks. You put 875,000 aside and leave the rest. The fourth win returns 900,000 Marks. You have won 900,000 + 875,000 + 104,600 - 10 = 1,879,590 Marks. And unlike the lottery, they are all yours, you need only share with your loved ones.

Or what about the horses? Say there is a full field of fifteen starters in a particular race. You go to the bookie, knowing nothing about horse-racing, and pick three of them to come in first, second and third (in that order, a so-called Trifecta bet). Arithmetically, the odds are 1/15 * 1/14 * 1/13 = 0.0003663 or one in 2,730. We'll arbitrarily worsen than by a factor of 1,000 to allow for the lamentable fact that there are good horses and bad horses. The odds against your selection winning are one in 2,730,000. Divide that into fourteen million and you get 5.1. Picking the winning three horses at random is five times more likely than winning the lottery. (And much more fun too.)

Do you still want to play the lottery?

If so, then at least get it right. Every lottery company has a website these days, you can [probably] find there a very useful piece of information: the list of least-frequently-chosen numbers (e.g. the number 7 is chosen twenty times more often than the number 13). If you make your selection from the list of least-frequently-chosen numbers, then the chances of winning are no worse or better than any other set of numbers — but if you do win, you are likely to be sharing the jackpot with fewer people. Your piece of the pie will be that much larger.

If you are interested in probability and risks, then you would probably ;-) enjoy reading Against the Gods, the remarkable story of risk by Peter L. Bernstein. The book is a history of mathematics, Bernstein examines the development of statistical and mathematical methods of calculating probabilities, and their effects on the evaluation of risks, and how this in turn lead to the development of modern financial analysis. He also describes the psychology of loss and gain: finding a twenty on the sidewalk would make most people very happy, but those Germans whose lottery jackpot was worth only 2,000 Marks were surely bitterly disappointed.

I found it fascinating.
Copyright © John Skinner, 2002. All rights reserved.
Thanks to John Amadio for correcting my math!
Last updated 2002.12.15