"How many hands do I need to play..."

[HesseJam]

You just need to replace NORMSINV and SQRT with the Portuguese names of the functions. You'll find the Portuguese name for NORMSINV in the streak sheet in column J (Amount Won (BB's, hour)). SQRT is "square root". You should find the Portuguese equivalent easily under your function ( fx ) button (look for math&trigonometry).

[ArthurD]

Do I input total hands played in B1, or is it sets of 100 hands played as Homer indicated? This makes a major difference in the outcome of the formula.. I'm using the formulas BisonBison reposted. Great formula, just wanna make sure the valeus I'm seeing are correct.

[Homer]

[ QUOTE ]
Do I input total hands played in B1, or is it sets of 100 hands played as Homer indicated? This makes a major difference in the outcome of the formula.. I'm using the formulas BisonBison reposted. Great formula, just wanna make sure the valeus I'm seeing are correct.

[/ QUOTE ]

If for win rate and standard deviation you are using BB/100, then for total hands you use sets of 100 hands. If you're using BB/hr, then for total hands you use hours played.

-- Homer

[ArthurD]

I have never done this before, but this thread is SO useful that it really needs to get bumped. I really think it anwsers a question that so many ask. Anyways, *BUMP*.
Sorry if this breaks any rules.

-Arthur

[stankybank]

[ QUOTE ]
Introduction

Recently, I've noticed a lot of players asking how many hands they need to play in order to prove that they are a winning player or that they are winning at a certain rate. Usually, the responses received are along to lines of:

- The vague answer -- "You need way more hands than that"

- The random number answer -- "You need at least 100,000,000 hands"

Anyway, I've decided to go over the process of computing confidence intervals, which will allow you to make mathematically valid statements about your true win rate.

Details

In order to compute a confidence interval, you need to locate certain statistics in Poker Tracker, specifically total hands, total table-hours, EV/table-hr (Amount won / Hours played) and SD/hr. All of these statistics are located under Session Notes -> Session Summary, with the exception of standard deviation, which is displayed after a user clicks the "More Detail..." button.

I just started using Poker Tracker again to track my 5/10 SH play, so I'll use those numbers as an example since I was planning to compute the confidence interval for that data anyway.

Hands (h) = 3115
Table-hrs (n) = 32.45
EV/hr (ev) = $39.41
SD/hr (sd) = $182.97

Once you have these statistics, you can use the below formulas to compute the upper bound (u) and lower bound (l) for a given confidence interval (c).

u = ev - normsinv[(1-c)/2]*sd*[1/sqrt(n)]

l = ev + normsinv[(1-c)/2]*sd*[1/sqrt(n)]

For the sample data I set the confidence interval at 95% and with these formulas found that the upper bound is $102 and the lower bound is -$24.

With this information, I can make the following statement:

"I am 95% confident that my true win rate is between -$24 and $102 per table-hr."

-- If I change the confidence interval to...

75%; Lower bound = $2 Upper bound = $76

50%; Lower bound = $18, Upper bound = $61

-- If I keep the confidence interval at 95% and change the hours played to...

100 (9600 hands); Lower bound = $4, Upper Bound = $75

250 (24,000 hands); Lower bound = $17, Upper bound = $62

1000 (96,000 hands); Lower bound = $28, Upper bound = $51

5000 (480,000 hands); Lower bound = $34, Upper bound = $44

10,000 (960,000 hands) Lower bound = $36, Upper bound = $43

From this, you can see that after 3000 hands you can't prove much of anything. It is within the realm of possibility that I have a true win rate of -2.5 BB/hr despite the fact that over 3000 hands I won at a clip of 4 BB/hr. You can also see that even after 500,000 hands, it is not a rarity for a player's true win rate to be 0.5 BB/hr more or less than his results to date.

Creating an Excel spreadsheet

It is quite easy to setup an Excel spreadsheet to compute confidence intervals. Information need only be entered in a few cells, as follows:

Cell A1 - "Hours Played"; Cell B1 - Hours played value (from PT)

Cell A2 - "EV/hr"; Cell B2 - EV/hr value (from PT)

Cell A3 - "SD/hr"; Cell B3 - SD/hr value (from PT)

Cell A4 - "Confidence Interval"; Cell B4 - ".95"

Cell A5 - "Upper Bound"; Cell B5 - <font color="green">=A2-normsinv((1-A4)/2)*A3*(1/sqrt(A1))</font>

Cell A6 - "Lower Bound"; Cell B6 - <font color="green">=A2+normsinv((1-A4)/2)*A3*(1/sqrt(A1))</font>

After you do this, you can mess around with cells A1, A2 and A4 (probably not A3 as your SD/hr converges fairly quickly and isn't likely to change unless your playing style does).

-- Homer

[/ QUOTE ]

[IsaacW]

[ QUOTE ]
What are the limitations on this formula?

[/ QUOTE ]
The formula as shown needs the true SD to be accurate. We do not have that information, but we do have the sample SD (the number that PT reports). When using the sample SD, you need to use something called the t-distribution to find confidence intervals that are wide enough. Using the normal distribution will result in smaller confidence intervals than the true values for small samples.

Information about the t-distribution.

[La Brujita]

Isaac,

I looked at the link but it didn't make a lot of sense to me (I am dumb). Are you saying that if my confidence intervals are say +/- .5bb/100 as determined by PT stats, t-distribution would indicate larger intervals? And generally how much larger?

Thanks in advance.

[Baulucky]

This post is so good that merits a bump.

[Nomad84]

I thought Student's T Distribution only applied when there were a small number of samples. Don't remember the cutoff point, but if you have enough sessions, this shouldn't matter, should it? How many is "enough"? 30? 40? 50?

[mmbt0ne]

I've done quite a few t-distribution problems, and never heard of this "enough" number you speak of. Are you sure you're not confusing something else, like normal approximation of the binomial?