## Thursday, October 10, 2013

### Even more fun with infinite lotteries

If you have an infinite fair lottery, it's possible to boost the probability that some member of a group wins without boosting the probability of any particular member of that group winning.

For suppose you have a genuine lottery where the tickets are numbered 1,2,3,..., and then a winning ticket is picked fairly. Presumably, the probability that an even-numbered ticket will be picked by the organizers is 1/2. And the probability that a ticket whose number is divisible by four (4,8,12,...) will be picked is 1/4.

Now suppose that after all the infinitely many players have bought their tickets, a mad ticket swapper goes around in the middle of the night. She takes all the tickets and redistributes them so that all the people with ticket numbers divisible by four get even ticket numbers, and all the people with ticket numbers not divisible by four get odd ticket numbers, with every ticket number being had by somebody. We can make the swapping rule be this if we like:

• 1→1,2→3,3→5,4→2,5→7,6→9,7→11,8→4,9→13,....
So what did the ticket swapper do? She doubled the chance that someone from the set of people whose original ticket numbers were divisible by four would win. For the probability that someone from that set would win without the swapper doing her thing was 1/4, i.e., the probability that a number divisible by four would be picked. But after the swap, the probability that someone from that set would win is 1/2, i.e., the probability that an even number would be picked.

But the mad ticket swapper did not change anybody's chances at winning. Let's say you started out with ticket 8. You now have ticket 4. If the lottery was fair, the probability that 4 is picked is exactly the same as the probability that 8 is picked. So while the probability that someone from your group would win has gone up from 1/4 to 1/2, this makes no difference to your personal probability of winning—-or to anyone else's!

Note that the above argument works no matter whether one says individual winning probabilities are zero or infinitesimal.

You might draw from the above the conclusion that there can't be a countably infinite fair lottery. While the conclusion would be correct, the inference might be a mistake. For you can do this with darts, too. See the next post.