How to Win at Craps: Variance
If the casino has such an advantage over you, why in the world does anyone play craps at all? My guess is that most
people don't have a clue they're playing a losing game. Others are so arrogant they think they can outplay the casino and
turn a negative expectation into a positive. Knowledgeable players know they'll lose over time, but they play anyway for fun
and excitement. As a knowledgeable player, why should you even bother playing a game that you know you'll lose? As a
knowledgeable player, is there any hope you can walk away a winner even though you're at a statistical disadvantage?

Casino craps is a game of statistics, with the casino having a built-in advantage. Since craps is based on statistics, we need
to find a way to use statistics to our advantage. You'll never beat the casino over time, but you can beat it in the moments
when the distribution hiccups and things go your way.

"Variance" is the average squared deviation of each number from the mean of a data set. Huh? What? Don't worry. We
don't need an MIT calculus degree to understand this. "Variance" is simply a measure of how spread out the data is. To
illustrate this, let's consider the familiar coin-flip example.

Suppose we flip a coin 10,000 times. We expect tails to appear about 5,000 times and heads to appear about 5,000 times.
Suppose we bet $1 on heads for each coin flip. If these are even-money bets, we expect to break even after those 10,000
coin flips. As noted in another lesson, the casino doesn't give us even money when it loses. In our coin-flip example,
instead of paying us $1 for each loss, suppose the casino pays us only $0.96. With this built-in casino advantage, our
negative expectation is to lose about $200 after 10,000 flips. The math is as follows. After 10,000 flips, if we expect about
5,000 tails and about 5,000 heads to appear, then we expect to lose 5000 x $1 = $5000; and we expect to win 5000 x
$0.96 = $4800. $5000 - $4800 = $200. So, we expect to lose $200 over 10,000 coin flips if the casino only pays us $0.96
each time we win, but we have to pay the casino $1 each time the casino wins. This is called "negative expectation."

Now, let's throw "variance" into the equation. Of those 10,000 flips, suppose we focus on only 30 of them, and we keep
betting on heads. Of those 30 flips, we might see heads 25 times and tails only 5 times. This data fluctuation shows, for a
limited number of flips over a short period of time, we can get lucky and experience Nirvana where things go our way. I call
it a "Nirvana hiccup" in the distribution that causes a relatively high variance. In this example of only 30 flips (remember,
we're focusing on only a very small number of flips over time), we win $24 for the 25 heads (i.e., 25 x $0.96 = $24), and lose
$5 for the 5 tails (i.e., 5 x $1 = $5), which gives us a net win of $19. This short-term variance temporarily removes the
long-term negative expectation, which means there are, indeed, times when we can win.

Although you'll lose over time, there are times when you'll win because of variance. Suppose you take a three day vacation
in Vegas once a year and play four 1-hour craps sessions each day (i.e., a total of 12 hours for the trip). You could
conceivably get extremely lucky and hit that Nirvana hiccup during each session, and then go home a big winner. In that
case, you go home thinking you're a genius, a craps god, invincible, a world-class gambling stud. Yeah, okay. Don't quit
your day job.

Now, suppose you're a Vegas local who plays an hour every day after work. In this case, it's clear that whatever few
Nirvana hiccups you experience will be adjusted over time so you'll lose your shirt over time.

Therefore, the infrequent craps player can, indeed, consistently win if she's lucky enough to hit those Nirvana hiccups.
However, the frequent long term player has no chance of coming out a winner at the end of his craps life. Part of being a
smart craps player is knowing how to be around for those occasional Nirvana hiccups where the dice fall your way.

If you don't want to lose your shirt, you must understand the truth behind the math. Don't fall for bogus winning systems or
silly dice control claims. Distribution variance is the only thing that makes you a short-term winner. Nothing else. No
ridiculous dice-control technique. No bogus winning craps strategy. It's the distribution variance and nothing else. Got it?
Now, be smart and play smart.