In this note, I will pose a puzzle about a lottery involving an exponential number of people. It is my version of a probability paradox called "The Shooting Room", which was invented by John A. Leslie in connection with the Doomsday Argument. I will first explain the rules of the lottery and ask whether you should buy a ticket. I will then explain two different ways of thinking about the problem and ask which, if either, is correct.
First, assume that every person is assigned a unique number at birth which doesn't change throughout the person's lifetime. We'll call this number the person's SSN (social security number). Assume that every person knows his/her own SSN.
Assume that the population grows without bound; that is, there's no specific limit to how large the population can grow. (Let's suppose that humans have become spacefaring and spread out throughout the universe, while still managing to maintain to assign unique SSNs to everyone.)
The lottery has 6 phases:
- The lottery commission secretly rolls a fair pair of 6-sided dice until they come up snake eyes (probability 1/36.) Let R be the number of rolls it took, including the final snake-eyes. If snake eyes are never rolled, then the lottery never starts.
- The lottery commission waits until the population is at least 10^R (10 to the power R).
- The lottery commission makes a list of every person alive in order from lowest SSN to highest.
- The lottery commission informs the first 10^R people on the list that they are eligible to play.
- Each eligible player now decides whether to buy a single ticket. This decision must be made in isolation; players may not talk about the lottery. A ticket costs $1.
- The first 10^(R-1) people on the list are potential winners. The lottery commission pays $2 to every potential winner who bought a ticket.
Question: Suppose you have SSN 5055305732 (or whatever), you know the rules of the lottery, and you have been notified that you are eligible to play. Should you buy a ticket? What is the expected value?
Here are two ways of thinking about the problem.
On the one hand, (number of potential winners) / (number of people eligible to play) = 10%. Thus, only 10% of the people who are eligible to buy a ticket would be winners if they did so. So you shouldn't buy a ticket.
On the other hand, you effectively became eligible to play on one of the R dice rolls: the first 10 people on the list were automatically eligible, then 90 more people became eligible on the first roll if it wasn't snake eyes, then 900 more people became eligible on the second roll if it also wasn't snake eyes, and so on. At each roll, the probability of a later group of people, approximately 10x larger, becoming eligible is 35/36. If this happens, the people who were already eligible will be in the first 10%. Thus, the probability that you are a potential winner, given that you are eligible to play, is about 35/36. So you should buy a ticket.
Which line of reasoning, if either, is correct?