[BruceZ]

This is what I was saying about the 1-sided confidence interval. You will be +/- 1.96 standard deviations from the mean 95% of the time as you say, but you will be LESS THAN 1.96 standard deviations ABOVE the mean 97.5% of the time, and you will be GREATER THAN 1.96 standard deviations BELOW the mean 97.5% of the time. We are only ignoring one tail of the bell curve instead of both tails. You will be less than 1.64 standard deviations above the mean 95% of the time, and greater than 1.64 standard deviations below the mean 95% of the time. BUT...THIS IS DOES NOT MEAN YOU HAVE A 5% RISK OF GOING BROKE AND THIS IS A COMMON AND TERRIBLE ERROR!!! I don't know how Mason says to do risk of ruin because I haven't read this book (though it will arrive in a few days).

What is true is that if you play for the number of hours for which performing 1.64 standard deviations below average would mean you lost your starting bankroll, you have a 5% chance of being broke at that point. This is not your total risk of going broke, because you may go broke before you ever get to that point unless you find some more money so you can keep playing. Assuming you have a way to keep playing, the chance of losing your starting bankroll and needing more money before you play for any number of hours can be shown to always be more than twice the risk of being broke once you play for that number of hours. Also, the amount of money represented by 1.64 standard deviations changes over time. If you want the time at which 1.64 standard deviation below average represents a maximum loss, maximize 1.64sigma*sqrt(n) - nu. The maximum occurs at n = (1.64sigma/2u)^2 hours. While you have a 5% chance of being down 1.64*sigma dollars at this point if you play to this point, it turns out you have over a 10% chance of being down that amount of money before this point (or any other point).

In a recent post, I calculated that you need a 172 bet bankroll for a 10% risk of ruin with an hourly rate of 1.05 bets and a sigma of 13.11 bets/hr. See this post for the correct formula. If I were to use the wrong method, I would say I need to be no more than 1.28 standard deviations below the mean, and the number of hours for which this is the largest number of bets is [1.28*13.11/(2*1.05)]^2 = 63.9 hours. Then since my average win at that time is 63.9*1.05 = 67 bets, and 1.28 standard deviations is 1.28*13.11*sqrt(63.9) = 134 bets, you would conclude I only need 134-67 = 67 bets to have a 10% risk of ruin, when in fact I need 172 bets, over 2.5 times as much! If I only had 67 bets, my risk of ruin would be (.1)^(67/172) = 40%, 4 times as great as I thought! Don't do this.