tojedobro: Q: Calc avg total wagered to reach goal, no time constraint

[tojedobro]

Any help would be appreciated. I am interested in determining the average expected aggregate sum wagered if I reach a goal before going bankrupt without a time constraint. (Not considering any amount wagered if I do go broke.)

In particular, I would like to calculate the aggregate amount in units I expect to wager if I succeed in reaching my goal, G, before going bankrupt (without consideration of amounts wagered if I go broke and without a time constraint) and consistent with Formula 3 from Ch. 8 of BJAIII: probability of reaching a goal (G) = e^[mu (G-B)/sigma^2] times [sinh(mu/sigma^2 * B) / sinh(mu/sigma^2 * G)]; where mu=per hand ev in units, B=initial bankroll in units, G= my goal > B in units, sigma^2=variance per hand in units.

Is it sufficient to simply sove the following equation for exptected total units wagered to answer the question: advantage = mu / 1 unit = (G-B) / exptected average total units wagered.

Thank you in advance.

[Don Schlesinger]

Let me preface my answer by saying that I am not at all sure if I am correct. However, it seems to me that ...

Since you don't care about the times that you go bust, you are really simply asking, how long in hours, on average, it will take you to reach a certain win goal, because, once that is known, you simply multiply by the average bet size, and you know how many units you have bet to accomplish your goal.

Now, when you do succeed, you will do so in an array of hours that should be normally distributed: Sometimes you'll get lucky and reach the goal very quickly; other times, it will take much longer than "normal." A problem may occur because you are truncating the distribution, in effect chopping off all the times when you fail, and I'm not sure if that affects the answer.

In any event, if it doesn't, then you simply need the mean of the distribution for the times that you succeed, and I don't know why that wouldn't be simply the amount needed to be won to reach the goal, divided by the hourly win rate at which you are playing. Once you know that, you multiply by average bet size, and that should produce your answer.

But, don't quote me on it! :-)

Don