Now you're asking a different question again.
Like I explained above, this notion of your winrate after n hands being more reliable (when asking the question, how likely is it that the actual value is greater than 0)... this does NOT mean the variance is less.
This is because the variance increases proportionally to the square root of n, whereas winrate is based on BR, which increases at a constant rate.
The variance is always increasing, but it increases at an ever-slower rate as n increases.
BUT the winrate represents a constant (unchanging) slope - a straight line.
The winrate is constant, but the variance ... whatever the slope ... increases at a slower rate.
So the winrate is climbing up past the 0 axis steaily. At first, the variance increases at a very fast rate... but then it slows down.
Here:
The bottom graph is the short term, and the top graph is the long term.
These are graphs of a 4.3bb/100hands winrate @ 80bb/1hand variance
These are typical values for an mediocre to good TAG player, beating the micros.
The red and blue lines show the upper and lower bounds for a 95% Confidence Interval.
The dotted green line is EV(bankroll), and the slope of that line is the winrate.
The green line looks curved in the upper (long term) graph because the x-axis is distorted to a logarithmic scale to better show the data.
The variance dominates in the short term. However, at ~30,000 hands, you can see that the constant slope of the winrate has equaled out with the ever-decreasing slope of the variance. (That's where there's a valley in the blue line in the top graph.) After that, the variance stops expanding in the negative, and since the slope is always less than the winrate's slope, the winrate dominates from that point on. At ~120,000 hands, you can see that the lower bound of the 95% CI crosses the x-axis, indicating a <5% chance of being a losing player at that point.
At no time does the variance shrink. Just the portion which falls below the 0 mark on the x-axis (Profit).
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Also note that the green line represents the center of a distribution that is symmetrical above and below that point for that value. Since the green line is always further from the x-axis as n increases, the portion of the distribution which falls below the x-axis is always less of a %-age of the distribution for that value of n.
So, even though the variance is increasing at all times, even in the short term, the fact that the winrate is positive means that the probability of being a losing player is always less with increasing n.
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