Stardate 20010824.0743 (On Screen): There's generally a lot of contempt in more intellectual circles for the lottery; it's sometimes called a "stupidity tax", for instance. And for the most part it is true that heavy participation in the lottery is less than wise. I recall that once a couple in Massachusetts sold their home and spent everything they got from it to buy lottery tickets. As one would expect, they got back about 60% of what they'd spent. They'd made enough separate wagers that "regression to the mean" had set in and their return was very close to the average return for the lottery as a whole. Part of the attraction of the lottery is due to the fact that most people have at least some understanding of big quantities of money, but no equivalent understanding of very low odds.
Is a long-odds bet worth taking? That's actually been something that game theorists dealt with a long time ago, and the way of determining it is actually quite easy. It requires you to calculate what's known as the "expected payoff" for the bet, which is the mean winnings. If, for instance, I have a 10% chance of winning $100, then I multiply the total winnings ($100) by the probability (0.1) yielding an expected payoff of $10. Equally, one chance in 100,000 of winning a million dollars has an expected payoff of $10. That number is then compared against the cost of playing the game. If the expected payoff exceeds the cost of playing then it is a "good bet". If it is less then it is a "bad bet". If it is the same then it is a "fair game".
Ordinarily the lottery is a very bad bet. Most state lotteries return something under 60% of their take as winnings, which means that the expected payoff on a $1 bet would be less than $0.60. But accumulating jackpots alter the situation somewhat. If there is a jackpot that doesn't get won, it rolls over into the next try. Does this make it become a good bet? Well, maybe. Suppose that a $50 million jackpot rolls over, and this inspires $200 million of ticket sales. The state keeps 40% of the new money, or $80 million, and returns $120 million into the pool along with that $50 million, for a total of $170 million. Since $200 million was wagered to win $170 million, the expected payoff has risen from $0.60 on a $1 bet to $0.85 -- better, but still a bad bet. The lottery actually can become a "good bet" but only if the rollover is larger than the amount of money the state will pocket from new bets in the next round. So if the state is keeping 40%, then if the rolled over jackpot is more than 40% of the amount you expect everyone to wager the next time then it's worth playing. A rolled over $50 million would be worth playing for if the next round drew $100 million in betting, because the expected payoff on a $1 bet would be $1.10. Unfortunately, that means that what you're really wagering on is crowd psychology, not a random draw. The thing to do is to wait until the last instant and make the calculation. (discussion in progress)
