Posted by David D'Aquin on June 27, 1997 at 08:41:49:I don't know whether I've found something new here, maybe just a rediscovery, but I have found the way to find the confidence intervals for N trials.
Some people define the long run as the amount of time needed for the expected win to exceed three standard deviations. When such is the case for fixed bet sizing the number of bets needed can found by:
Long run= [(SD per bet * 3) / exp win per bet]^2
How does this relate to the number of bets needed for average log growth to overcome a situation where the number of wins is a certain number of SDs to the left? It turns out that the confidence is exactly one half that for fixed bets for any size N. If we calculate the number of bets needed for fixed bets to overcome 2 standard deviations to the left, then this is the exact number of hands need for Kelly betting to overcome 1 standard deviation to the left. So for Kelly betting to find the same level of confidence for the long run the formula would be:
Long run= [(SD per bet * 6) / exp win per bet]^2This, by the way, is for perfect Kelly betting... not interval resizing as we use in blackjack. But there is a similar formula for that also. SBA gives all the stats needed to calculate it. I'll post a specific example later.
- Confidence levels for interval resizing David D'Aquin 13:21:58 6/27/97 (1)
- Last formula, a different kind of Long Run David D'Aquin 11:44:27 6/28/97 (0)