Risk of ruin question

[BruceZ]

[ QUOTE ]
The main problem is that you estimated the probability that you would be down 20 BB at the end of the period, but we want the probability that you bust out at some point, which is significantly higher.

[/ QUOTE ]

In fact it is always at least twice the endpoint value.

This post gives a short-term risk of ruin formula intended to compute a "vacation" bankroll, as the OP is looking for. This is from Blackjack Attack by Don Schlesinger. Do not use the other method of doubling the endpoint value as that is only accurate when the EV is small compared to the bankroll; however, it does provide a lower bound on the risk of ruin.

Here are some results from this formula for EV = 1 bb/hr, SD = 10 bb for 1 hour, and a trip bankroll of 20 bb. These are rounded to the nearest percent. I wouldn't put alot of stock in these numbers for such short intervals for the reason that Pzhon mentioned. I give them here mainly so that you can confirm your understanding of the formula. Then you can apply it to somewhat longer intervals. Schlesinger gives a blackjack example for 16 hours at 100 hands/hr, and he says that the formula should err on the conservative side.

1 hour: 4%
2 hours: 13%
4 hours: 26%
8 hours: 39%

[uclaben]

Bruce - thanks, this is great. I messed with the Schlesinger formula in Excel to model a number of different scenarios. If anyone is interested, the results are below - variables are win rate, starting bankroll, and number of hours played. I left SD at 10bb/hr. If anyone wants me to reproduce the numbers at a different level of variance, I'd be happy to do so.

Thanks again.

winrate=1bb/hr

starting br
# of hours
20 50 100
1 3.7% 0.0% 0.0%
2 12.8% 0.0% 0.0%
4 25.7% 0.7% 0.0%
8 38.6% 4.5% 0.0%
12 45.2% 8.7% 0.1%
15 48.4% 11.4% 0.3%
20 52.1% 15.1% 0.9%
30 56.4% 20.4% 2.2%
50 60.6% 26.3% 4.9%


winrate=1.5

starting br
# of hours
20 50 100
1 3.3% 0.0% 0.0%
2 11.5% 0.0% 0.0%
4 23.0% 0.6% 0.0%
8 34.2% 3.4% 0.0%
12 39.8% 6.4% 0.1%
15 42.5% 8.4% 0.2%
20 45.5% 11.0% 0.5%
30 48.9% 14.5% 1.2%
50 51.9% 18.1% 2.5%


winrate=2bb/hr

starting br
# of hours
20 50 100
1 3.0% 0.0% 0.0%
2 10.3% 0.0% 0.0%
4 20.4% 0.4% 0.0%
8 30.1% 2.5% 0.0%
12 34.7% 4.7% 0.0%
15 36.9% 6.0% 0.1%
20 39.2% 7.8% 0.3%
30 41.7% 10.0% 0.6%
50 43.6% 12.0% 1.1%