RoR in X amount of hands.

[ShakeZula06]

Let's say I'm bankrolled with 250 BBs, and will play X hands to clear a bonus at a table/site/limit where I have 1.5 BBs/100 hand WR and a SD of 17/100 hands. What formula would I use to calculate my RoR?

I'm experimenting with some bonus whoring and since I've had my money spread around in a few places at no game do I have the recommended 400-500 BBs for limit 6max at any particular room. However since my moneys only on there for a certain amount of hands I figure my RoR is much smaller. Does that sound right?

[ShakeZula06]

Nobody? Well I was thinking about it.

Let's say you have to play 5,000 hands to clear the bonus, which would mean for you to not clear it your WR would have to be -5 BB/100 hands for you to not comlete it.

SO then the question becomes, what are the chances that a player with a WR of 1.5 BBs/100 hands and a sd of 17 BBs/100 hands runs at -5BBs/100 hands or worse in 5,000 hands.

Now that I think about it I think I have to find the chance that I will never be down 250 BBs during any block of X hands.

Thoughts? Links? Anybody?

Thanks,
Ben

[BruceZ]

[ QUOTE ]
Nobody? Well I was thinking about it.

Let's say you have to play 5,000 hands to clear the bonus, which would mean for you to not clear it your WR would have to be -5 BB/100 hands for you to not comlete it.

SO then the question becomes, what are the chances that a player with a WR of 1.5 BBs/100 hands and a sd of 17 BBs/100 hands runs at -5BBs/100 hands or worse in 5,000 hands.

Now that I think about it I think I have to find the chance that I will never be down 250 BBs during any block of X hands.

Thoughts? Links? Anybody?

Thanks,
Ben

[/ QUOTE ]

As a rule of thumb, your risk of going broke sometime during the 5000 hand interval is a little more than twice the probability of being broke at the end of the 5000 hand interval, assuming that you always play that long (see note below). On average you should be up 75 BB at the end of the 5000 hands, so being down 250 BB is 325 BB below average. The standard deviation for 5000 hands is 17*sqrt(50), so the probability of being down 325/(17*sqrt(50)) standard deviations is given by the Excel function =NORMSDIST(-325/(17*SQRT(50))) =~ 0.34%. So your probability of going broke sometime in the 5000 hands interval is a little more than twice this or a little more than 0.68%. The short term risk of ruin formula from Blackjack Attack by Don Schlesinger gives 0.89%.

NORMSDIST[(-250-1.5*50)/(17*SQRT(50))] +
EXP[-2*1.5*50*250/(17^2*50)]*NORMSDIST[(-250+1.5*50)/(17*SQRT(50))] =~ 0.89%.

So both methods agree that your risk of ruin is a little less than 1%.

EDIT: Fixed typo in first method. Also, this first method (rule of thumb) cannot be used for a general number of hands, and it is only a reasonable approximation in this case because 5000 hands is sufficiently small. To be specific, the number of hands must be much less than 100*[2.7*17/(2*1.5)]^2 =~ 23,400, which is the number of hands for which being 0.34% =~ 2.7 standard deviation below average gives a maximum loss.

[ryanj247]

[ QUOTE ]
[ QUOTE ]
Nobody? Well I was thinking about it.

Let's say you have to play 5,000 hands to clear the bonus, which would mean for you to not clear it your WR would have to be -5 BB/100 hands for you to not comlete it.

SO then the question becomes, what are the chances that a player with a WR of 1.5 BBs/100 hands and a sd of 17 BBs/100 hands runs at -5BBs/100 hands or worse in 5,000 hands.

Now that I think about it I think I have to find the chance that I will never be down 250 BBs during any block of X hands.

Thoughts? Links? Anybody?

Thanks,
Ben

[/ QUOTE ]

As a rule of thumb, your risk of going broke sometime during the 5000 hand interval is a little more than twice the probability of being broke at the end of the 5000 hand interval, assuming that you always play that long. On average you should be up 75 BB at the end of the 5000 hands, so being down 250 BB is 325 BB below average. The standard deviation for 5000 hands is 17*sqrt(50), so the probability of being down 325/(17*sqrt(50)) standard deviations is given by the Excel function =NORMSDIST(350/(17*SQRT(50))) =~ 0.34%. So your probability of going broke sometime in the 5000 hands interval is a little more than twice this or a little more than 0.68%. The short term risk of ruin formula from Blackjack Attack by Don Schlesinger gives 0.89%.

NORMSDIST[(-250-1.5*50)/(17*SQRT(50))] +
EXP[-2*1.5*50*250/(17^2*50)]*NORMSDIST[(-250+1.5*50)/(17*SQRT(50))] =~ 0.89%.

So both methods agree that your risk of ruin is a little less than 1%.

[/ QUOTE ]


bruce, i'm having some problems here. i'm probably just tired and missing something obvious...

when i run the first formula in excel, i get a different result (i get 99.8%). to get .34%, i have to change "350" to "-325". "-325" seems more logical to me...is that just a typo??

i went ahead with the assumption that it was. i was able to run the 2nd (schlesinger) formula in excel and get the same result of ~.89%.

then i wanted to see what would happen if we kept everything the same except the number of hands. instead of 5000 hands, i wanted to use 50,000 hands.

at the end of 50,000 hands, you should be up 750BB. so being down 250BB is 1000BB below average. so the excel function should be NORMSDIST(-1000/(17*SQRT(500)))*2 = ~.85%. sounds about right.

i tried using the same scenario for the schlesinger formula. it ended up looking like this:

NORMSDIST((-250-1.5*500)/(17*SQRT(500))) + EXP(-2*1.5*500*250/(17^2*500))*NORMSDIST((-250+1.5*500)/(17*SQRT(500))) = ~7.19%.

did i make a mistake somewhere? why such a big difference between the two formulas for a larger number of hands?

[lewfather]

Okay, I have a question.... I found a ROR calculator but how do I find my SD?

[BruceZ]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Nobody? Well I was thinking about it.

Let's say you have to play 5,000 hands to clear the bonus, which would mean for you to not clear it your WR would have to be -5 BB/100 hands for you to not comlete it.

SO then the question becomes, what are the chances that a player with a WR of 1.5 BBs/100 hands and a sd of 17 BBs/100 hands runs at -5BBs/100 hands or worse in 5,000 hands.

Now that I think about it I think I have to find the chance that I will never be down 250 BBs during any block of X hands.

Thoughts? Links? Anybody?

Thanks,
Ben

[/ QUOTE ]

As a rule of thumb, your risk of going broke sometime during the 5000 hand interval is a little more than twice the probability of being broke at the end of the 5000 hand interval, assuming that you always play that long. On average you should be up 75 BB at the end of the 5000 hands, so being down 250 BB is 325 BB below average. The standard deviation for 5000 hands is 17*sqrt(50), so the probability of being down 325/(17*sqrt(50)) standard deviations is given by the Excel function =NORMSDIST(350/(17*SQRT(50))) =~ 0.34%. So your probability of going broke sometime in the 5000 hands interval is a little more than twice this or a little more than 0.68%. The short term risk of ruin formula from Blackjack Attack by Don Schlesinger gives 0.89%.

NORMSDIST[(-250-1.5*50)/(17*SQRT(50))] +
EXP[-2*1.5*50*250/(17^2*50)]*NORMSDIST[(-250+1.5*50)/(17*SQRT(50))] =~ 0.89%.

So both methods agree that your risk of ruin is a little less than 1%.

[/ QUOTE ]


bruce, i'm having some problems here. i'm probably just tired and missing something obvious...

when i run the first formula in excel, i get a different result (i get 99.8%). to get .34%, i have to change "350" to "-325". "-325" seems more logical to me...is that just a typo??

[/ QUOTE ]

Yes. I fixed the typo in the original post. Note that this was the method I said not to use in general, as it only works when the number of hands is sufficiently small. I made a note of this in the original post also.


[ QUOTE ]
i went ahead with the assumption that it was. i was able to run the 2nd (schlesinger) formula in excel and get the same result of ~.89%.

then i wanted to see what would happen if we kept everything the same except the number of hands. instead of 5000 hands, i wanted to use 50,000 hands.

at the end of 50,000 hands, you should be up 750BB. so being down 250BB is 1000BB below average. so the excel function should be NORMSDIST(-1000/(17*SQRT(500)))*2 = ~.85%. sounds about right.

i tried using the same scenario for the schlesinger formula. it ended up looking like this:

NORMSDIST((-250-1.5*500)/(17*SQRT(500))) + EXP(-2*1.5*500*250/(17^2*500))*NORMSDIST((-250+1.5*500)/(17*SQRT(500))) = ~7.19%.

did i make a mistake somewhere? why such a big difference between the two formulas for a larger number of hands?

[/ QUOTE ]

The 7.19% is correct. Note that when the number of hands is this large, the "short-term" risk of ruin approaches the risk of ruin for playing forever which is exp(-2*1.5*250/17^2) =~ 7.5%. This is because you are unlikely to go broke after you have made it to this point, because your bankroll will have grown by a factor of 3 on average, which raises your risk of ruin to the 3rd power (making it negligible). The first method which gave 0.85% is no longer valid because the number of hands is too large to apply this endpoint approximation.

[BruceZ]

[ QUOTE ]
Okay, I have a question.... I found a ROR calculator but how do I find my SD?

[/ QUOTE ]

You can get it from Poker Tracker software www.pokertracker.com, or compute it yourself using the formula in this post. Note that you can replace hours with the number of hands, or units of 100 hands. You can find an example worked out in the book Gambling Theory and Other Topics by Mason Malmuth.

[ryanj247]

ok, that makes sense. thank you.

[BillC]

Does Schlesinger's trip ROR formula have a derivation in print. If so, I'd like to see it.

[BillC]

I've been looking for a derivation for the trip ROR formula. Can anyone point me in the right direction?
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