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A ruin question5 September 2008
I recently received the following email from one of my readers:
Well, Padme you certainly don't ask easy questions. Let me first address an issue in your first question. You can't have both a 4 to 1 chance of success and a 5% risk of ruin. I suspect that you are mixing the answers to two different questions here, so let's ignore the 5% figure for the 500 hands.
The classic ruin problem is a one-dimensional random walk with a unit step in either direction. What makes the ruin question in Blackjack so difficult is that because of splits, doubles, and 3-to-2 payoffs for Blackjack the size of the step becomes random. I don't know how to handle a random walk with a random step size. If I were going to try to seriously tackle this question I would probably create a simulation of some sort and crunch the numbers on a computer. That said, there is an approximation that one can make that will give you "ball park" figures.
Blackjack, when played using basic strategy, has a house edge of around a half percent. So, we can create a simple random walk with a house edge of 0.005 and see what the ruin probability looks like for such a game. If p represents the probability of success and q the probability of a loss then p - q represents the player's expected return. By our assumption
p - q = -0.005
and since q = 1 - p
2p - 1 = -0.005
Solving for p we see that p = 0.4975. Thus q = 0.5025. For the classic ruin problem we need the ratio q/p (which I'll call r) and it is 1.10050251. For your first question, rather than use 500 playing for 600, I'll use 50 playing for 60 with a unit step. The classic ruin calculation is then r raised to the 60th power minus r raised to the 50th power all divided by the quantity r raised to the 60th power minus 1. This number, representing the ruin probability, turns out to be approximately 0.21. Hence your 4 to 1 figure looks about right.
For the 40 question the ruin probability is approximately 0.094, which represents a chance of success of about 11 to 1. For the 30 question the ruin probability is approximately 0.072 so the success rate is about 14 to 1. I'll leave the 20 question for you Padme. See you all next month.
Don Catlin can be reached at email@example.com
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