Distribution Variance : Our Weapon Against the Casinos

I n our article about house advantage, we learned how the casino gets its edge over the player by paying us casino odds instead of true odds.  You think, “Why would anyone play and fight a losing battle if the casino has an edge that’s going to make them lose?”  Most people don’t understand their mathematical disadvantage, or don’t even know the house has an advantage.  They’re the clueless saps who drop $200 on the table, lose it in an hour, and the walk away wondering why they’re so unlucky.  Some people are so desperate to make easy money that they get suckered into believing that wacky betting systems and silly dice-rolling techniques (i.e., “dice control”) can actually alter the house advantage either by reducing it or by turning it into a player advantage.  Instead of asking why you should play a game that’s stacked against you, the real question you should ask is if the game will allow you to win or at least break even during your occasional Vegas vacation?

varianceCraps is entirely about math and statistics.  The game is designed with the math stacked against you.  Knowing that fact, the key is to uncover a method of using the math and statistics against the casino.  Before continuing, you must accept the fact that you won’t defeat the casino over a long period of time.  The key words are “over the long term.”  For example, if you play eight hours every day for a year, then at the end of that year, you’ll have lost your shirt.  So, don’t think you can take what you learn from our articles and quit your job to become a professional craps player.  Don’t do it because you’ll lose all your money.  However, if you play only a couple of hours a week at your favorite online casino, or if you play eight hours a day during your four-day Vegas vacation, then you definitely can defeat the casino if the statistical distribution gets out of whack causing the outcome to fall in your favor instead of the casino’s.

The key, and the only weapon against the casino, is the phenomenon called “distribution variance.”  Don’t worry, you don’t need a PhD in statistics to comprehend it.  Variance measures how spread out a bunch of data is.  Let’s look at an example, a coin flip, to make it easy to understand.

flippingcoinLet’s flip a fair coin at least 10,000 times.  There are only two possible outcomes for each flip, either heads or tails.  Therefore, of those 10,000 flips, heads should show half of the time (5,000) and tails should show the other half of the time (5,000).  Let’s bet $1 that heads will show for each of the 10,000 flips.  If the house gives us even money on those bets, then we should break even when all flips have been completed.  In other words, when heads appears 5,000 times, we win $5,000.  When tails appears 5,000 times, we lose $5,000.  In the end, we expect to break even.  But we know from our other articles that the house doesn’t pay us “true odds” when they lose.  Instead, they pay us “casino odds” so they can make a profit (casino odds are a little less than true odds).  For example, suppose the casino doesn’t pay us the full $1 when they lose a flip.  Suppose they only pay us $0.96.  It’s 4 cents less than the dollar they should pay us, but it’s only 4 cents so we can live with it, right?  The house has a built-in advantage.  When we lose a flip, we have to pay $1.  But when we win a flip, we only get $0.96.  Therefore, with expect to lose $200 after all the flips have been completed.  Here’s the math.  Assume the distribution variance breaks even after 10,000 flips, which means for 5,000 flips heads will show, and for the other 5,000 flips tails will show.  Also assume that we bet on heads each flip.  For the 5,000 flips where tails shows and we lose, we pay the casino $5,000.  For the 5,000 flips where heads shows and we win, the casino pays us only $4,800 ($0.96 multiplied by 5,000 flips equal $4,800).  Therefore, our net loss after 10,000 flips is $200 if the casino only pays us $0.96 instead of the full $1 for the 5,000 bets that we win.

Before continuing, let’s see if we can figure out the house advantage for this example.  If you need to review our other article about calculating house advantage, then do it now.

For 10,000 flips and a $1 bet for each flip, out total bet investment is $10,000 (i.e., 10,000 flips x $1 bet each flip = $10,000).  As shown above, after 10,000 flips, if we lose $5,000 and win $4,800, our net loss is $200.  Dividing our net loss of $200 by our total investment of $10,000, we see that the house advantage for this example is 2% (i.e., $200 / $10,000 = 0.02).  See how easy it is to calculate the house advantage?  Now, let’s get back to variance.

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Remember our 10,000 coin flips?  Let’s take a small sample of the 10,000 and look at just 30 of those flips.  Remember, we’re betting on heads every flip.  Now, because we’re dealing with a much smaller sample size (i.e., 30 instead of 10,000), the chances are much greater that we’ll see the outcomes be more skewed to heads or tails.  For example, how many times have you flipped a coin 10 times and seen either heads or tails come up, say, 7 or 8 times instead of the 5 times that we expect?  Because we’re dealing with a much smaller sample size, there’s a much greater chance that the results can get out of whack; however, the results eventually even out as the sample size gets bigger.  When the results are out of whack (e.g., for 10 flips, when 8 heads appear instead of the 5 that we expect), that’s the distribution variance at work (i.e., the results vary from what we expect).  As noted, over a long period of time, the distribution variance is negligible because everything evens out after a long period of time or when the sample size is big, such as 10,000 flips.  But for a small sample size, the variance can be very large.  It is distribution variance that produces the familiar hot and cold streaks that we commonly experience at a craps table.  The casino doesn’t care about the short-lived hot and cold streaks because the casino knows they occur only during short periods of time, and eventually the distribution approaches equilibrium and evens out.  That’s the key for the casino; over time with hundreds of people playing day after day, month after month is when the casino achieves its massive profits.  Let’s get back our example of the coin flips.

Our small sample size of only 30 flips might show their outcomes as, for example, 25 heads and 5 tails.  Remember, variance measures how spread out a bunch of data is.  The 25 heads and 5 tails illustrate that the data spread can be greatly skewed over a relatively small sample size (i.e., in this case the small sample size is only 30 flips out of 10,000).  What this means to us at the craps table is that during a small sample size of time (e.g., one hour over three days), the variance can take a turn and produce a wicked hot streak where we defeat the casino and need a wheelbarrow to carry our chips away.  Going back to the coin flip, when heads shows those 25 times we win $24 (25 flips multiplied by $0.96 equal $24).  When tails shows those 5 times, we lose $5.  Our net overall win for those 30 flips is $19.  This brief disruption in the variance gives us a temporary chance to clobber the casino.  That, and only that, is how we beat the casino at craps.  No silly betting system, no wacky con-job known as “dice control,” none of that nonsense.  It’s when the distribution results get temporarily out of whack in our favor that allows us to have a net gain instead of a loss.

Although you’ll likely have a net loss if you play long enough, you can expect to win big occasionally because of distribution variance.  Let’s say you go to Las Vegas for three short days.  Suppose you plan to play four short one-hour periods each of the three days, which totals 12 hours of craps time.  It’s entirely possible that you could experience one of those disruptions in the variance that causes the temporary results to swing in your favor.  As a result, you’re a big winner for that short trip.  Now, in contrast, suppose you’re a local who likes to play four hours after work each day, day after day, week after week, and month after month.  In this case, the sample size of time is very large, which means the distribution variance has a chance to even out.  So, whatever winning hot streaks you experience over that long period of time will be cancelled out by all the losing cold streaks that you experience.  Your winning times and losing times will even out over a long period time.  But, in this case, with the built-in house advantage where the casino doesn’t pay their fair share to you when you win (i.e., they pay “casino” odds, not true odds), you’ll likely end up a loser after a long period of time.

You, as a casual occasional player, truly can win if you’re lucky and experience the disruptions in the distribution variance.  But if you’re hooked and play frequently, your chances of winning are severely hurt.  To be successful, you must have the discipline to limit your play.  Don’t play eight hours a day every day.  Instead, make a date with your spouse or better half to play together at your favorite online casino maybe once every two or three weeks for a couple of hours.  Although the casino has a built-in advantage even in a perfect distribution, in the relatively short time you play (i.e., a small sample size), you might hit a variance in the distribution, an anomaly, where the odds seem to take a vacation and favor you instead of the casino.  When that time comes, everything you do is right.  The air is pure, the sun is shining, there’s peace on Earth, and your bankroll skyrockets.

So, how do we apply this wonderful phenomenon called “distribution variance” to our game of craps so we can beat the casino?  That, my friend, is the subject of more articles.  You’re doing a great job!  Keep reading our articles and keep learning.  The more you read and learn, the better–and more successful–player you’ll be.

You can now head over to the table of contents to find more great content.

Written by John Nelsen in partnership with the team of craps pros at crapspit.org.