## Devlin's Angle |

June 2000

There were two winners, one from Illinois, the other from Michigan. They shared the prize equally: $181.5 million each. During the period from the March 7 drawing through the May 9 drawing, Big Game ticket sales in all participating states totaled more than $565 million. Illinois alone sold over $100 million worth of tickets over the life of the jackpot.

The lottery organization makes no secret of your chances of winning. According to their website, the odds against winning the jackpot are around 76 million to 1. The website doesn't explain how they arrive at that figure, but it's an easy calculation.

The game requires that you choose 5 different numbers, each between 1 and 50, plus a sixth number between 1 and 36. To win the grand prize, your five numbers have to agree with the five numbers selected by the lottery computer (the order does not matter) and your sixth number has to be the same as the sixth number chosen by the computer. Thus, your odds of winning are:

1 in [50x49x48x47x46] / [5x4x3x2] x 36which works out at 1 in 76,275,360.

Presumably that sixth number is there -- with a range of 1 to 36 -- so that the odds work out at around 70 million to 1, a figure that, given the expected number of entries, will guarantee a jackpot winner roughly every two to four weeks. In this way, the lottery organizers can keep the level of interest high.

The math is easy. The hard part it getting your mind around the answer. Just how can you get a sense of odds of such magnitude?

Roughly speaking, those odds are slightly longer than throwing heads on 26 successive tosses of a fair coin. (That unlikely event has odds of 1 in 67,108,864.) Given that most people would be reluctant to bet on getting 5 heads in a row, let alone 26, this comparison makes it clear that the psychology of lotteries has a logic all of its own.

Actually, when it comes to coin tossing, many people's intuition leads them to an erroneous conclusion that, in a sense, goes against the lottery comparison. Seeing a run of 3 or 4 heads in a row, they believe that the odds of getting a tail are increased -- that "it's time for a tail to come up." The longer the run of heads, they believe, the more likely it becomes that you'll get a tail. Not so. This is known as the Gambler's Fallacy. The coin has no memory. The odds of getting a head or a tail on any one throw remain exactly one-half no matter how many previous throws have resulted in a head. (Assuming the coin is a fair one.)

What other ways might we have of trying to appreciate the
odds against winning the Big Game? Well, imagine laying
standard playing cards end to end from New York to San
Francisco. The underside of just *one* of those
cards is marked. Start to drive across country, and at some
point stop and pick up a card. If you've chosen the marked
card, you win the jackpot. Chose any other card and you
lose. How much would you be willing to pay to play this
game? In terms of the odds, you've just played the Big
Game.

Or imagine a standard NFL football field, which has a playing area measuring 100 yards by 55 yards. Somewhere in the field, a student has placed a single, small, common variety of ant that she has marked with a spot of yellow paint. You walk onto the field, blindfolded, and push a pin into the ground. If your pin pierces the marked ant, you win. Otherwise you lose. Want to give it a go? If you do, then in terms of the odds, you'll be playing the Big Game.

Of course, there is a sure fire way to ensure you win the Big Game. For a stake of $76,275,360, you can buy tickets that cover all possible combinations of numbers, and one of them would be sure to win. With a jackpot of $325 million, this looks like a rock solid way to make a massive profit. There's still a small risk of losing, however. If four or more other people also pick the winning combination, then you all share the prize equally, and you lose money.

The real problem with this approach, however -- leaving aside the small fact that you need $76 million to start with -- is the time it would take to buy the tickets. If you were to choose numbers at an average rate of one per second, taking no breaks, and were to work like this 24 hours a day, 7 days a week, all year round, it would take you 15 years to cover all possible combinations.

However you look at it, the odds against winning the Big
Game jackpot are truly staggering. Does that mean that the
best strategy is not to play at all? Oddly enough, the optimal
strategy is to play, but to restrict your wager to an amount of
money that is truly of *no* value to you.

For most adults in the United States, $1 really is the same as
nothing. If they lose a dollar bill, they think nothing of it. Now,
if you don't enter the lottery, your chances of winning are
absolutely zero. You will never win. Ever. If, on the other
hand, you buy a single dollar ticket, you have a small, *but
non zero* probability of winning. That's not merely
"slightly" better than having no chance at all; it's in an
entirely different category. (As any sophomore mathematics
major knows -- or should know -- a small, nonzero epsilon is
*very* different from setting epsilon equal to zero.)

Now take into account the fact that many people gain considerable enjoyment from the anticipation of waiting for the lottery draw -- of imagining what it would be like to have all that money -- and it's not at all hard to understand why lotteries like the Big Game are so popular. Indeed, the excitement of playing may well be worth the price of a $1 ticket.

What's that you say? Have I tried my hand at the Big Game? No. The truth is, I've never bought a state lottery ticket in my life. Yes, I know I just argued that the optimal strategy is to enter with a small stake. But that fact is, I just can't get myself past those odds.

Devlin's Angle is updated at the beginning of each month.