James on Gambling

Statistics 101 for Gamblers
-by James

There is no need to be a mathematician to gamble in a casino, but a little mathematical knowledge is all that is needed to make you a realistic, if not a winning, gambler.

If you gamble long enough in a casino, you will lose all of your money. The simple reason is that the casino has constructed the games so that they make money (on average) on every bet you make. For example, there are 38 numbers on a roulette wheel, but the casino pays as if there were only 36. In the long run, the casino keeps $2 of every $38 wagered at double-zero roulette. There are a few expert players who can actually profit in the long run in the casino, for exactly the reverse reason. An expert video poker player playing full pay Deuces Wild will (over millions of hands) win 7 cents of every $10 she wagers. Expert blackjack players counting cards in a deeply dealt game, and disciplined players receiving generous casino comps, can also profit in the long run. For online players, the GameMaster's directory at will tell you the casino's advantage (or player's advantage) on most of the games you will find online.

In the "short run", however, all of these players can and will have winning and losing nights, weeks, or even months. Gambling is a risky undertaking. The casino may average a 5 cent profit on every $10 bet at blackjack, but if the casino always won exactly 5 cents per hand, no one would play. The risk inherent in gambling can be measured by a quantity called the standard deviation. If your gambling results are "normally" distributed, then 67% of the time your bankroll will end up within one standard deviation above or below your average ("expected") result. Two standard deviations contain 95% of our results, three standard deviations contain 99.7% of all outcomes. The word "normal" here is a technical word referring to the normal curve (bell curve) in statistics. However, we can gloss over this technicality due to a mathematical fact called the "law of large numbers" which in our case says that if you make enough wagers, your bankroll will follow a normal distribution. For games with a relatively even distribution of outcomes (like blackjack) "enough" may be just 100 bets, but for jackpot games with rare high payout events like video poker, the long run can be quite long indeed. A rough rule of thumb might require that you play long enough to hit the rare jackpot (on average) 5-10 times to consider yourself in "the long run".

I said that the GameMaster's directory listed the casino advantage for most online games, but not the standard deviation. Fortunately, the standard deviation is easy to calculate for most games. The variance of a game is the average squared result of your wager. For craps with no odds, this is trivial; you always win one bet or lose one bet, and the variance is 1. If you make column bets in American roulette, in 38 spins you will win 2 bets 12 times, and lose one bet 26 times. The variance is 1/38 x (12 x 4 + 26 x 1) = 1.95. (To be precise, we should subtract off the square of the casino advantage, but this correction should be negligible if you are playing any reasonable game.) The standard deviation of the game is simply the square root of the variance, for instance the standard deviation of the column bet at roulette above is 1.4 bets.

What can we do with these numbers? Assume that we take 100 spins betting $10 on the first column at double zero roulette. On average, we expect to lose 5.26% of our $1000 in bets, or about $50. The standard deviation of our 100 spins is found by multiplying the game's standard deviation (1.4 bets from above) by the square root of the number of spins (10) to get a standard deviation of $140. Thus while our probable results will be centered around $53, 95% of our results will fall between a win of $230 and a loss of $330. If you lose more than $470 (three standard deviations below the average), you have hit a horrible string of luck that should happen only about 1 time in 1000, and you may be justifiably concerned about the fairness of the game.

If your game is blackjack, the standard deviation per hand is roughly 1.1 bets. The number is slightly larger than one because splits, doubles, and blackjacks all give you the opportunity to win or lose more than one bet per hand. Here is a recent example, unfortunately drawn from real life. I played 135 hands of blackjack at Casino Bar when they offered a 35% deposit bonus to try their new software. The rules here are horrendous (D10 only in multi-deck), and I will assume a theoretical casino edge of 1%. Playing 135 hands, I expected to lose an average of 1.4 bets. The square root of 135 times 1.1 gives a standard deviation of 12.8 bets. A little more than one time in 1000 I could expect to lose about 40 bets. My actual loss was 60 bets, or 4.6 standard deviations below the expected result. This should happen only 5 times in a million sessions in a fair game; for practical purposes, this is proof that I was cheated by their new software, whether by bugs or intentionally.

Now that you know the casino advantage and standard deviation for your favorite game, what should you do with these numbers? One thing for online players is to monitor your actual results compared to your expected outcome. While a casino could cheat in a subtle way and avoid detection, if you see results from 100 or more hands that are outside of three standard deviations from your expectation, you should be suspicious. If other players you trust have reasonable results, it's possible that you are indeed that one person in 1000 who will have such horrible luck, but use caution.

More generally, these numbers can give you a realistic idea of the risks inherent in gambling. If you like to play blackjack at $10 a hand, at 200 hands per hour for four hours at a time, your standard deviation is about $300. You could easily be behind by $600 after a session, or more if you are seriously unlucky. If that number gives you cause for concern, you are betting too high. Of course, you could just as easily win $600, and if you can accept this level of risk, go ahead and place your bets, knowing you are gambling responsibly within your means.

Send any comments or suggestions to me at

And until next time, may your results exceed your expectations.

The GameMaster, Living The Good Life

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