Sampling With/Without Replacement.
When we play poker and we decide to bet, raise, re raise, shove, we are gambling or betting on an outcome. The cards are shuffled, dealt and we bet. We then bet the flop, turn and river and await the result. Then, the dealer collects all cards and starts over. We do it all again, and again and again. This is called sampling with replacement.
You what [yes, I can hear you thinking it] wtf is sampling without replacement, never friggin heard of it. Neither had I until recently. But I decided to share it anyway. Hopefully the reasons why will become apparent.
Say you have an urn with 55 green marbles and 45 red marbles [deck of cards]. You are making even money bets on green, say $1. Someone closes his eyes, selects a marble, settles up the bet and sets the marble aside. Then you repeat the same thing 99 more times. After you're done, you're guaranteed 55 wins and 45 losses, for a net gain of $10. This is called sampling without replacement. Unfortunately, in the world of poker, and pretty much all gambling, it doesn't work this way.
Oh, you still use the same urn, the same 55 green marbles and the same 45 red marbles [deck of cards], you're still betting $1 on green, but after a marble is selected and the wager is settled, the marble is replaced in the urn. This is called sampling with replacement. Because the marble is replaced, the odds
are always 55/45 in your favor on every selection. However, here the result after 100 times is not guaranteed. The best probability theory can say is that if you repeat the wager a large enough number of times, your actual results will converge towards the predicted results, which are what you are guaranteed in the first example.
Simply put, variance is how you get to where you're going to end up. But in the real world, it can take any number of paths. Just as you could pick out 10 red marbles in a row at any given time, you could also pick out 10 green ones. Eventually, you'll pick out 55% green and 45% red, but you never get there in a nice, neat, tidy series. Strangely, no one has ever given an explanation for why it works out this way, it just does.
So the next time you are affected by variance, try and think of this and you might understand just what is going on and that in itself will help you deal with variance.