Probability of losing X BBs

[Fianchetto]

If I have a known win rate and std dev. how do I calculate the probability of losing X BB's in a session?

For instance, say I have a win rate of 1.5BB/HR, and std dev. of 14.7BB/HR, what is the chance I would lose 70BB in an hour.

What if my session is 2 hours?

Thanks.

[topspin]

[ QUOTE ]
If I have a known win rate and std dev. how do I calculate the probability of losing X BB's in a session?

For instance, say I have a win rate of 1.5BB/HR, and std dev. of 14.7BB/HR, what is the chance I would lose 70BB in an hour.

[/ QUOTE ]

I'll take a stab at this.

If you assume that your results are normally distributed (i.e. bell-curved), then you first need to figure out how many standard distributions your swing is. In your example, 70BB is 70/14.7 = 4.76 standard deviations. (Note that since 95% of the time you'll be within 2 standard deviations, your 4.76 SD swing is going to be pretty unlikely.)

You can calculate the exact probability using the Q-function. In Excel, you'd calculate Q(x) by entering 0.5*ERFC(x/SQRT(2)). In your case, it's a 10^-6 probability.

[ QUOTE ]
What if my session is 2 hours?

[/ QUOTE ]

In a 2-hour session, your standard deviation would be 14.7/sqrt(2), so 70BB is 4.76*sqrt(2) = 6.73 SD's. The probability of exceeding that is Q(6.73) = less than 10^-11.

[Nottom]

[ QUOTE ]
In a 2-hour session, your standard deviation would be 14.7/sqrt(2), so 70BB is 4.76*sqrt(2) = 6.73 SD's. The probability of exceeding that is Q(6.73) = less than 10^-11.

[/ QUOTE ]

You math here is fudged up. Your STd Deviation over 2hours should be larger than for 1, not smaller. So your StdDev is 14.7*SQRT(2) or 20.8 and you would be on a 73/20.8 = 3.51 SD downswing. (this is 73 becasue you expect to make 3BB and instead lost 70 so the diff is 73))

I don't have the ERFC function available to me becasue its not installed, so I can't do the rest of the math for you, but I would guess its somewhere around 1% likely to happen.

[topspin]

[ QUOTE ]
[ QUOTE ]
In a 2-hour session, your standard deviation would be 14.7/sqrt(2), so 70BB is 4.76*sqrt(2) = 6.73 SD's. The probability of exceeding that is Q(6.73) = less than 10^-11.

[/ QUOTE ]

You math here is fudged up. Your STd Deviation over 2hours should be larger than for 1, not smaller. So your StdDev is 14.7*SQRT(2) or 20.8 and you would be on a 73/20.8 = 3.51 SD downswing. (this is 73 becasue you expect to make 3BB and instead lost 70 so the diff is 73))

I don't have the ERFC function available to me becasue its not installed, so I can't do the rest of the math for you, but I would guess its somewhere around 1% likely to happen.

[/ QUOTE ]

My bad, you're right. For the curious, Q(3.51) works out to 0.000222807, or about 0.02%.

[Fianchetto]

Thank you, I think this is coming back to me now, been awhile since I took statistics.

So if I express my win rate and std. dev. in terms of BB/100, lets say

WR = 2BB/100 and Std Dev. = 17.45BB/100

and I want to know what is the chance that I would lose 70BB in 300 hands, I would just take

17.45* SQRT(3) = 30.22 = my std. dev. for 300 hands.

so 76/30.22 = 2.51

then plug in 2.51 into the Excel .5*ERFC(2.51/SQRT(2))

and I get .00603 or about a 1 in 165 chance of losing 70BB in 300 hands.

Does that seem right?

[Nottom]

Statistically, yeah it seems right. Problem is that except over the very long term, poker just doesn't really follow a normal distribution.

Look at your typical 300 hand session for example. Most of the hands played are going to win 0BBs becaseu you fold them preflop. Of those you play, most of them will be either a small loss or a small win. Those few hands that are leftover are usually the ones that define how your session goes. Did you make all your draws and drag the big pots or did you see suckout after suckout with all your big hands? Since these hands are relatively rare (you might have 4-6 truely big hands in a typical session) its easy to win or lose all of them in a given session and have a much bigger win or loss than you would expect via a calculation.

[MiloBloom72]

[ QUOTE ]
Since these hands are relatively rare (you might have 4-6 truely big hands in a typical session) its easy to win or lose all of them in a given session and have a much bigger win or loss than you would expect via a calculation.

[/ QUOTE ]
So does the standard deviation tell us anything useful about how we're playing? (Except as, you say, over the really long term.)

[Nottom]

A low one should still indicate that you are likely a bit passive and a high one will normally indicate looseness or aggression, but I don't think it really tells you a whole lot about your game.