It's occurred to me that the standing joke that 'it's always 50/50' (i.e. some n00bs can't comprehend that probability applies to card games because 'it either happens or it doesn't') actually contains some truth, when projecting forwards. Looking back and knowing what the cards were it's nonsense of course, but at time X0, the probability that a player will be ahead or behind at times X1, X2, X3 etc. will stay pretty close to 50% for a while, and maybe for a long time.

I think it would be interesting to see what this looks like for a player with known win rate and standard deviation. Intuitively I imagine the graph will look like a sine wave, starting and ending fairly flat with a steeper incline in the middle. It would be interesting too to know the midpoint where the upward gradient is maximal. To be clear, I'm talking about a graph where the x-axis is time/number of sessions/multiples of 100 hands, and the y-axis is the probability of being winner.

I realise this requires some form of calculus but I can't remember how to do it. Help please!? Also what free software can I use to graph this?

Thanks

Bumpity

Don't make me have to get out my copy of Gambling Theory... and try to figure this out for myself!

So you're saying, you want a graph where
f(x) = chance that player is net positive after X hands?
You could write a little program to do it, or use excel. I assume you know how to calculate f(x) for fixed values of X? If not I can post the formula or you can check out my calculator, here's an example:

http://rustybrooks.com/poker/stats.h....x=&outside.x=

That's for a player with win rate of 2bb/100, stdev of 15bb/100, after 1000 hands (chances of being a net winner are about 66%)

Actually I could modify this script to get the results you want. I just popped into work but maybe if I have a few minutes later.

OK, check this script out:
http://rustybrooks.com/poker/stats2.html
you enter in a win rate, standard deviation, and a # of hands range in the format of
0:1000:100
which is from 0 hands to 1000 hands, in intervals of 100

It spits out some CSV which is # of hands, and pct chance of being ahead after that amt of time. Copy/paste the CSV into excel and graph it and you're done.

OMG Rusty you are the nuts

will post some results if I get anything interesting

http://www.castrovalva.com/~la/win.htm

Thanks Alex, that looks like a sweet tool as well.

It looks like I was wrong in my initial hypothesis about the graph taking a while to get away from horizontal. Graphing some results from Rusty's app:



Here are some results showing how many hands are needed to be ahead for various win rates in NLH and PLO. I've used poker tracker big blinds/100 and standard deviations of 50 and 70 respectively which I think are fairly standard.