Monday, August 03, 2015

A Thought Experiment / Puzzle to Entertain You

Suppose the U.S. government instituted a new lottery for all citizens that worked like this:
  1. On the first day of every month, each person who wants to play produces a single penny, which they will use to play the game.
  2. Each player flips his/her coin once, with one of the following results: 
    • Tails: The player is eliminated from the game, and the penny he/she was using is paid as the cost of participating.
    • Heads: The player is still in the game and continues to the next round.
  3. Repeat (2) until:
    • There is only one player left, who is the lottery winner, OR
    • There are no players left (because all remaining players flipped tails in one round), and there is no winner for the month.
  4. If there is a winner, that person gets the total amount paid by all players.  In other words, the prize money is calculated in dollars as: (total number of participants) / 100.
As an example, suppose an average of 300 million people play each month (about 25 million fewer than the total U.S. population).  Each person pays a penny, so the players will pay a total of $3 million to play. If one person wins, that person will win the full $3 million, and the lottery will generate no income for the government.

However, on months when there is no winner, the government will take in the full $3 million as income with no pay-out.

The cost to the government of administering this lottery is $1 million per month.

Here is the question: Assuming 300 million people play every month, will the government make money or lose money with this system? 

Assume that the pennies used all have a perfectly 50/50 chance of coming up heads or tails when flipped, and that there is no possibility of cheating by either the players or the government.