2play or not 2play?

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!

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