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

[Homer]

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