Average # of Bets to Bust & Chance of Reaching Target Formulas

[aka23]

I'm familiar with Don Schlesigner's risk of ruin formulas and have found them to work quite well. However, there are some related values I'd like to check, for which I have not been able to work out a formula.

I'd like to know the chance of being above a target gain after playing a fixed number of hands with a finite bankroll. So the play continues after reaching target, but stops if bankroll reaches 0. The result can be approximated with a cumulative Normal function, if risk of ruin is low, but this method is not accurate when risk of ruin is high. I'm looking for a formula that works in both cases.

I'd also like to measure average number of bets during the cases when bankroll reaches 0. For example, with a certain bankroll and number of hands, risk of ruin might be 20%. What is the average number of hands played during the 20% of the time when the player busts.

Does anyone have any ideas?

 

[Kasi]

Does anyone have any ideas?

I'm not exactly sure what you're asking and I'm not sure if I could help you if I was

Norm gave me a formula in the "Is it True" thread in response to a question I asked about achieving a target you may want to look at.

Are you maybe asking about the point at which maximum loss occurs and short-term ROR formulas give wrong results because you'd actually need more bankroll for fewer hours than the formula gives? Probably not, since this is covered by Don.

Then there's one I was just trying to understand in general and see if I could relate it to the maximum loss formula of Don's - when the number of trials passes the point of maximum short-term risk and the difference between long-term risk and shorter-term trip risk begins to narrow approaching zero at the limit. There is a formula for the point of max short-term risk in it.
It's very interesting in that it points out that the traditional thinking of how you bet when you spread to 2 hands is true for long term-risk calculations but not for shorter-term risk, like trips, when trials are relatively limited like they are likely to be on a trip.
It's over at BJmath.com and was written by Bryce Carlson and any counter wanting to spread on trips should give it a read. You can grasp the point without grasping the math. I'm proof of that lol.

But I think this is what accounts for the fact "if they don't get you early, they probably aren't going to get you at all".

 

[aka23]

Are you maybe asking about the point at which maximum loss occurs and short-term ROR formulas give wrong results because you'd actually need more bankroll for fewer hours than the formula gives? Probably not, since this is covered by Don.

Thanks for the links. They were interesting reading, but I don't think it's what I was looking for. I'm asking about two different events.

The first one is the chance of reaching a target gain with a limited bankroll. There are several existing formulas to do this, but the ones I have seen only treat the target as a barrier that ends wagering. I'm interested in chance of reaching the target if you continue wagering until you either reach the specified number of hands or bust.

The second one is the average number of bets until ruin. If your bankroll reaches zero 20% of the time, I'm interested in the average number of bets during that 20% of the time when you bust.