What's the formula for % chance of winning (being in profit any amount) after x amount of hands?
Assume flat betting with a set player edge
Im guessing its probably somewhat similar to the n0 formula
Reading page 21 and not understanding any kind of formula to determine this.
It says "determine what % of the SD the corresponding expected return represents" then find the area under the curve to the left of this figure.
Can you explain this in a pure mathematical formula? Like, how do i even make a curve?
I have EV of game and SD, isnt this all i need in order to find % win after x hands?
blueman,
Let's pick some numbers to see how to calculate your answer. Let's assume you're wonging in with a 2% edge (so at about +5) and flat-betting (so your S.D. is 1.15). If you play 100 rounds, what is the chance you'll be even or ahead?
Well, in 100 rounds, you expect to win 2 units. The total S.D. will be 1.15*?100 = 11.5 units.
If you end up even, your z-score is z = (0-2)/11.5 = -0.17. In general, z = (result - mean)/(standard deviation).
If you consult Table 8.9 on page 147 of BJA3 (or else use the Cumulative Distribution function in, say, Excel), you'll see that the CDF for -0.17 is 0.4325: this means you have a 43.25% chance of being behind. Therefore, you have a 56.75% chance of being even or ahead.
Hope this helps!
Dog Hand
Doghand, what is 1.15*?100 supposed to represent? Or maybe formatting is wonky?
Wouldn’t the SD be:
Var = 1.15^2 = 1.3225
Var = 1.3225 * 100 = 132.25
SD = 132.25^0.5 = 11.5 (units)
?
"Everyone wants to be rich, but nobody wants to work for it." -Ryan Howard [The Office]
The ? was supposed to be a square root sign. What you did above, while correct, is unnecessary. Why bother squaring, to get variance, only to undo the square by taking square root? Just take the square root of the number of rounds (sqrt 100 = 10) and multiply by the single-hand s.d.: 10 x 1.15 = 11.5.
Then proceed as Dog Hand did. Get the z-statistic and refer to the chart to get probability of being ahead after specified number of rounds.
Don
Ah, you’re right! First time for everything, heh? j/k
I’ve just always done it that way because....I don’t know why? Easier to keep it together in my head, I guess. Speaking of z-scores and whatnot, I figured out how to do the formula on wolframalpha and keep a bookmark to it on my phone. Change the (0.25) to whatever z-score you want.
https://www.wolframalpha.com/input/?...%5E2)%2F2)++du
Unless there’s an easier (but still accurate/precise?) formula?
"Everyone wants to be rich, but nobody wants to work for it." -Ryan Howard [The Office]
Unfortunately there is no simpler formula because the Gaussian normal distribution function cannot be analytically integrated because of the -u^2 square in the exponential function. Many functions can be analytically integrated by another function given as a short formula (the so-called anti-derivative function from Calculus), but not the Gaussian one.
So you need large tables with function values or a computer in order to approximately compute the integral (= area under the curve), e.g. by the Simpson integration rule, which requires about 10 to 100 iteration steps in a loop to get sufficient precision in the results, depending on the desired number of precise digits.
Last edited by PinkChip; 07-21-2019 at 05:39 AM.
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