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quasi
Posts:
10,726
Registered:
7/15/05


Re: A gambling paradox
Posted:
Jun 4, 2014 4:42 PM


Michael Press wrote: >Paul wrote: >> >>The following consideration appears somewhat paradoxical to >>me, in the sense of defying one's intuition. As far as I know, >>it's an original idea. >> >>Suppose that a casino offers all games at fair value (and >>doesn't have any other charges such as an entry fee etc.) >>For example, there are games where you have a 50% chance >>of winning an amount equal to your stake etc. Suppose also >>that all the customers are compulsive gamblers who always >>gamble everything they have until they lose all their money. >>My immediate intuition would be that, of course, the casino >>wins money  the customers are determined to throw their >>money away. However, that can't be true because for every >>transaction, the casino makes an expected gain of zero. >>So the casino will make zero profit in the long run no >>matter how hard the customers try to lose. I suppose it's >>a bit similar to the St. Petersburg paradox. Every once >>in a while you'd get a customer who keeps winning until >>the casino closes and that would wipe out the profits. > >A player determined to play as long as it takes walks into >a casino with a bankroll of n. The casino has a bankroll of >1000n. Most likely the player will loose all his money. In >fact, the odds against the player are 1000:1.
Change the numbers to:
player has 1 dollar
house has $100 million
Player cannot bet more than the house has left.
All bets offered by the house are at fair odds.
Player must play until he either wins all or loses all (no early exit is allowed).
Thus, when the game is over, the player has either won $100 million dollars or lost $1.
Since the individual bets are fair, it follows that regardless of the amount the player chooses to bet at each stage, the odds against the player ending up with a profit of $100 million is exactly $100 million to 1.
But those are _fair_ odds.
Thus, it's like a lottery, except unlike realworld lotteries which, by design, are very unfair for the player, this is a fair lottery.
People will be lining up to play  not just compulsive gamblers but ordinary people  the same ones that would typically buy a lottery ticket.
quasi



