Question for statisticians

[BruceZ]

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Supposing that one has a "true" win-rate of n bb/100. This is not actual win-rate but just an assumed number based on one's own (fluctuating) quality of play relative to the field.

Now, the law of large numbers says that your actual win-rate will approach n the more hands you play.

It seems like there should be some statistical formula involving standard deviation and number of hands played for just how fast actual win-rate (call it a) approachs n.

I would think this would actually be more like a probability function of the form: if n is your true win-rate and s is your standard deviation, and you've played h hands (in 100s), then the probability that a lies in the range [n-s/10, n+s/10] is p.


Anyone have a good formula (or formulas) for this?

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You want the standard error, which is simply s/sqrt(h), where s has units of bb for 100 hands. Then your actual win rate will lie within [n-s/sqrt(h), n+s/sqrt(h)] about 68% of the time, [n-2*s/sqrt(h), n+2*s/sqrt(h)] about 95% of the time, etc.

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OK, I'm taking a risk trying to correct Bruce here, but ...

let,
S1 = SD per hand
s = SD/100 hands
Sh = SD per "h" hands

general equation:

(Sh)^2 = h*S1^2

Therefore,

s^2 = 100*S1^2
(Sh)^2 = h*S1^2

so,

s^2/100 = (Sh)^2/h

Sh = sqrt(h)*s/10

if,

WR = your win rate/100 hands after "h" hands

then,

Total winnings = h*WR/100

95% confidence interval is,

2*Sh = 2*sqrt(h)*s/10

Total winnings with confidence interval is:

h*WR/100 +/- 2*sqrt(h)*s/10

multiply this by 100/h to get it in terms of per 100 hands,

WR +/- 2*s/sqrt(h/100) = WR +/- 20*s/sqrt(h)

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I'm not sure what you are "correcting". My 95% confidence interval is WR +/- 2*s/sqrt(h) not WR +/- 20*s/sqrt(h) because he stated that h is already in units of 100 hands.

If you are addressing the units of s, this is a common source of confusion caused by the way standard deviation is typically reported by Poker Tracker and others with units of "bb/100 hands". Technically variance would have units of bb^2/hand, so standard deviation would have units of bb/sqrt(hands), but what is meant is that they are evaluating the standard deviation in units of bb FOR a 100 hand interval, and thus giving you the result of bb/sqrt(hands) * sqrt(100 hands). I always state this as bb FOR 100 hands to make sure people use the correct numbers while still being technically accurate.

[Silent A]

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I'm not sure what you are "correcting". My 95% confidence interval is WR +/- 2*s/sqrt(h) not WR +/- 20*s/sqrt(h) because he stated that h is already in units of 100 hands.

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Oops, I didn't see that part. I never imagined someone would express it this way.

[donkeykong2]

i havent read all the responses but i make an intuitive approach:
if you double the sample size and divide the result by 2 your standard deviation will be 1/2^0.5*(the former standarddeviation)
ig you double the sample size twice the new standard deviation will be 1/4^0.5=1/2*(the former standarddeviation)
so you double your accuracy by taking a 4x bigger sample size.