SD as it relates to session results.

[MarkD]

Let's take a hypothetical player who we have accurate data on for a total of 1,000,000 hours. It doesn't matter how we got this data let's assume it exists.

For this 1,000,000 hours the player has a WR of 1 BB/hr and a SD of 10 BB/hr^(1/2).

From this we know his actual win rate is (within 99%):
WR = 1 +/- 0.03 BB/hr.

But in any given 8 hour session how much can he expect to win or lose?

My guess at the moment would be:

Win = WR*t +/- (3*SD/sqrt(t))*t
= WR * t +/- (3 * SD * sqrt(t))
where:
WR = win rate
t = time in hours
SD = standard deviation (BB / hr^(1/2))

which would give us a range for win/loss per 8 hour stretch of:
Win/loss = 8 bb's +/- 84.85 BB's.

Is this correct or am I way off?

[bigpooch]

The 100%x(1-alpha) confidence interval for t hours of play
would be

WR x t +/- (z(alpha)*SD)*sqrt(t)

where z(alpha) is the value of z that satisfies
phi(z)= 1-(alpha/2). Here, phi is the integral of the
standardized normal distribution function or

phi(x) = [integral from -infinity to x of] d(t)
where d(t) = (1/sqrt(2*pi))*exp(-(t^2)/2).

With z=3, this is about a 99.75% confidence interval; i.e.,
there should be only 25 sessions out of 10,000 that you
would outside of this. [Note phi(3.0)=0.9987]

A reasonable choice of z would be that for which alpha=0.01
so that z(0.01) = 2.58 so that a 99% confidence interval is
obtained. Then the range for an 8 hour stretch is "merely"
8 BBs +/- 72.97 BBs or [-64.97, 80.97]. This shows how big
the swings can be during only 8 hours of play.

On the other hand, after about a year of play, things look
rosy! After 2000 hours (equivalent of full-time B&M play),
the 99% confidence interval would be 2000 +/- 1153.8, so it
would be quite difficult for this hypothetical player to
lose during a year! Also, for those that are multitabling
with a similar hourly rate per table, it's not hard to see
that it is almost impossible to have a losing year playing
four tables.

[MarkD]

Perfect. It has been a while since I took stats during my engineering degree (and have never used it outside poker) but your integral rang a lot of bells.

And 2.58 rings a lot of bells too. Thanks for the very complete answer, and for confirming my fuzzy thinking.