Re: Probability of going broke
This is not correct, although it may provide a reasonable approximation.
The problem is, we are not interested in the simple sum of the trials, because it is possible to have bad runs (which bring the bankroll negative) followed by good runs, which bring the overall sum over n back to something we would expect.
That is, if X_i are the independent Bernouilli trials, the distribution of the sum of the X_i (call it S_n, which follow a Binomial distribution) is not sufficient to answer our question. We need the distribution of the minimum of the following sequence:
X_1, X_1 + X_2, ... , S_n.
Quite a different animal indeed. As I said though, I've forgotten how you solve for this distribution.
gm
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