Raffle Tickets
Marilyn vos Savant is a smart lady with some clever answers to puzzles in Parade Magazine. She gets the answers right nearly all the time! Uncharacteristically, she flubbed part of the answer to a "Raffle Ticket" question on December 13, 2009.
The question posed: "Say you buy 100 tickets in a raffle with 10 prizes. Which gives you a better chance of winning: putting all of the tickets in one basket, or spreading them among all 10 baskets?"
Vos Savant suggests putting all the tickets into the basket with the fewest tickets if discernable, or into the basket for the least-desirable prize. The spirit of her answer suggests that the goal is to win something, that is, not win nothing. The answers would be different if the goal was maximizing value, which would focus on winning the most prizes, or the most valuable prize to the contestant (whose desires may differ from other contestants).
If the baskets are hidden and the prizes are of equal value, vos Savant suggests "it doesn't matter what you do". Actually, you should spread the tickets evenly, 10 in each basket, to maximize the chances of winning something. This minimizes the "fratricidal" effect of your tickets against each other.
The simplest case to understand is where everyone else has put the same number of tickets in each basket. Lets assume that there are 100 tickets in each basket. If you put all your 100 tickets in one basket, then you have a 50% chance of winning that prize, and a 50% chance of winning nothing.
If you put 10 tickets in each basket, then you have a 10/110 chance of winning each prize, and a 100/110 chance of losing each prize. The chance of losing all 10 prizes is (100/110) to the 10th power, or 38.6%, so your chance of winning at least one prize is 61.4%. Better odds. Further, you could win more than one prize - your average "yield" is 10 times 1/11 or 90.9%. You have a 22.9% chance of winning two or more prizes, and even a 5.5% chance of winning three or more.
If the other tickets are not evenly distributed, then things get more complicated, but the even distribution strategy looks even better. Let's say that there are 8 baskets with 100 tickets, one basket with 50, and one basket with 150. Focus on these two baskets; the odds remain the same for the others. In the 100/100 case, you had a 100/110 chance of losing each basket, or a 1-(100/110)*(100/110) or 17.4% chance of winning one or both. In the 50/150 case, you have a 1-(50/60)(150/160) or a 21.9% chance of winning one or both. Adding perturbations to each pair of baskets just increases your chances, so you do even better when all the baskets differ. The "spread your investment" strategy works even better, because it is more likely to find the holes in the investment strategy of others.
If your values differ from other people (and they usually do), then you should weight your "investments" towards the items that you think you want more than others. If you are a skier in a community of boaters, you should bet extra tickets on the ski boots instead of the life preserver. You maximize your chances of winning something you want if you bet on that, or bet against what other people want. The most likely outcome is better for everyone - wouldn't you rather that life preserver goes to save a boater's life, rather than waste valuable space in your garage?
This thought problem is very important - these are exactly the same strategies you should use when investing in stocks. If you have the skills and the knowledge, invest in stocks with the "emptiest basket"; that is, the fewest other investors competing for the highest value. If (like me) you don't have the skills and knowledge, spread your investments out, so that you have a better chance of some of your money landing in an emptier basket.