Bankroll Formula – Part 3 – Win Rates

We can use the Standard Deviation to find how far our various session results are away from the average (mean) of all session results using the following 3 relationships:

* Standard Deviation = The square root of Variance
* Variance = [(1 / Number of Sessions) x Y] - [((WR / hour) ^ 2 / (Number of sessions)) x Session Hours]
* Y = The sum of (Session 1 Profit / Loss ^ 2 / Session Hours) + ... (Session n Profit / Loss ^ 2 / Session Hours)

NOTE: The calculation below uses Profit / Loss achieved in individual sessions and hours played, but hands played can be used instead of hours played in the formula.

For example, you play 10 Sessions with the following Hours / Profit (Loss):

1 ................... 8 / $ 100
2 ................... 8 / ($ 300)
3 ................... 8 / $ 200
4 ................. 12 / $ 200
5 ................. 12 / ($ 100)
6 ................. 10 / $ 250
7 ................. 10 / $ 400
8 ................. 10 / $ 50
9 ................. 12 / $ 300
10 ............... 10 / $ 100

Total Sessions ............. 10
Total Profit ................ $ 1200
Total Hours . 100
Win Rate ................... $ 12 / hour

Y = (100 ^ 2/8) + (-300 ^ 2/8) + (200 ^ 2/8) + (200 ^ 2/12) + (-100 ^ 2/12) + (250 ^ 2/10 ) + (400 ^ 2/10) + (50 ^ 2/10) + (300 ^ 2/12) + (100 ^ 2/10)

= 10000/8 + 90000/8 + 40000/8 + 40000/12 + 10000/12 + 62500/10 + 160000/10 + 2500/10 + 90000/12 + 10000/10

= 1250 + 11250 + 5000 + 3333 + 833 + 6250 + 16000 + 250 + 7500 + 1000

= 52666

Variance = (1/10) (52666) - ((12 ^ 2/10) (100))

= 5267 - (14.4 x 100)

= 5267 - 1440

= 3827

SD = Square Root of 3827 = 62 (it is 61.86 rounded up)

You can work out your expected Hourly Profit range to 3 levels of Confidence, 68%, 95% and 99.7%, using the following formula (SD = Standard Deviation, SQRT = Square Root and HP = Hours Played):

* 68% Confidence Level = 1SD / SQRT of HP = 62/10 = 6.20
* 95% Confidence Level = 2SD / SQRT of HP = 124/10 = 12.40
* 99.7% Confidence Level = 3SD / SQRT of HP = 186/10 = 18.60

Therefore you can be (for example) 99.7% confident of the range of your hourly Win Rate being $ 12 + or - $ 18.60

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Source by Richard Sun