Would you play a game that had an 83 % chance of doubling our money and only a 17% chance of losing?
Suppose you are asked to play a simple betting game which has the following rules:
- All the participants line up for a sequential chance to play against the house (ooooohhhh, the House!). [Assumption 1: There is an unlimited source of players for this game.]
- The first person to bets $100 and rolls a die. If a 1 is rolled the person loses the $100 to the house. If a 2, 3, 4, 5, or 6 is rolled the player is paid $100 by the house. [Assumption 2: The house has really, really deep pockets.]
- For the next turn, if the last roll resulted in a player losing $100, then the next player (only one) is selected for the next turn. However, if the last turn resulted in a win, then the next two players are selected for a turn. Both payers bet $100, but only one die is rolled to determine both of their fates. Again, if a 1 is rolled, both lose $100 to the house. If any other number comes up, both players win $100.
- Continuing in this fashion, whenever a player or group of players loses (a 1 is rolled) the next turn starts over with a single player, and if a win occurs (a 2, 3, 4, 5, or 6 is rolled) then the number of people to play the next round is double the number of the round just completed.
Complicated? Here is an example:
- Player 1 (only one player) bets, a 4 is rolled, player 1 wins $100.
- Players 2 and 3 (two players) bet $100, a 6 is rolled, each wins $100.
- Players 4, 5, 6, and 7 (four players) bet $100, a 1 is rolled, each lose $100.
- Player 8 (only one player this time) bets $100, a 6 is rolled, player 8 wins $100.
- Player 9 and 10 bet $100, a 1 is rolled, both players lose $100.
- … and so on.
For each player, probability states the present die roll is independent of all prior rolls implying there is a 5 in 6 chance (~87%) of winning and a 1 in 6 (~17%) chance of losing. Great odds! Would you play?
Before you answer consider the previous game with tallies at each stage.
- Player 1 (only one player) bets, a 4 is rolled, player 1 wins $100.
- House is down $100
- Players 2 and 3 (two players) bet $100, a 6 is rolled, each wins $100.
- House is down $300
- Players 4, 5, 6, and 7 (four players) bet $100, a 1 is rolled, each lose $100.
- House is up $100
- Player 8 (only one player this time) bets $100, a 6 is rolled, player 8 wins $100.
- House is even at $0
- Player 9 and 10 bet $100, a 1 is rolled, both players lose $100.
- House is at $200
- … and so on.
The point here is that each time a 1 is rolled, the house comes out an extra $100 dollars ahead. As the game continues, the house becomes richer and richer.
The questions you have to answer are ‘If a person’s chance of winning is 5 out of 6, how come the house is getting richer and richer?’ and ‘If the house is getting richer and richer, should I really play this game?’
When you can provide compelling answers, please let me know. To play or not to play has me stumped!