Let's say I have two bags, each with an infinite number of marbles.
In the "red" bag, exactly 2/3 are red, the rest are blue.
In the "blue" bag, exactly 1/3 are red, the rest are blue.
Let's say I draw 1 marble from each bag and keep a running average the number of red marbles in each.
Suppose after the first draw, I have:
Red bag: 0% red, 1 draw (got a blue)
Blue bag: 100% red, 1 draw (got a red)
After 1 marble, blue (100%) > red (0%), which is obviously wrong since we know the real distribution. However, if we didn't know the actual distribution, we couldn't say for sure we were wrong --- the probability that blue > red is not 0%.
What I am trying to do is to figure out when I can stop drawing marbles. That is, I want to keep drawing marbles until I am 99.9% certain that mean_red > mean_blue. (Or, if I get really unlucky, 99.9% certain that mean_red < mean_blue!)