risk of ruin over fixed sample size - Gambling and Probability

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risk of ruin over fixed sample size

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05-27-2008 , 09:15 PM

dsteff

enthusiast

Join Date: Oct 2006 Posts: 75

I've been looking through "The Mathematics of Poker" and have found and understand the formulas for calculating risk of ruin given a bankroll size, winrate, and variance. However these formulas are assuming the game is played for an infinite time.

My question is what if I'm only going to play a given limit for a certain number of hands (perhaps a weekend or month)?

Put simply I would like to calculate the risk of ruin given a bankroll size, winrate, variance, and sample size. For example, what would be the risk of ruin of a 200bb bankroll with a winrate of 1bb/100, a standard deviation of 20bb/100, and a sample size of 10000 hands?

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05-27-2008 , 09:33 PM

AaronBrown

Pooh-Bah

Join Date: May 2005 Posts: 4,240

Over 10,000 hands, you expect to win 100 bb with a standard deviation of 200 bb. So to be down 200 bb at the end means you were 1.5 standard deviations below your expectation, which has probability 6.68% or 1 / 15.

However, that doesn't answer your question. If you had an infinite bankroll, it's possible you could have lost more than 200 bb at some point in the 10,000 hands, but then come back to have a loss less than 200 bb by the end. Computing the probability of this is difficult, but it cannot do more than double the probability. So your risk of ruin must be less than 13.36% or 2 / 15. In this case, it's not a lot less.

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05-28-2008 , 04:54 AM

BruceZ

Carpal \'Tunnel

Join Date: Sep 2002 Posts: 11,877

Quote:

Originally Posted by AaronBrown

Computing the probability of this is difficult, but it cannot do more than double the probability. So your risk of ruin must be less than 13.36% or 2 / 15. In this case, it's not a lot less.

That is incorrect; it must be AT LEAST double for a winning player. What you said would be correct for a blackjack player against a house edge.

See this post for a short-term risk of ruin formula. Use the formula rather than the rule of thumb mentioned in that post unless the discussed conditions are satisfied.

Last edited by BruceZ; 05-28-2008 at 05:18 AM.

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