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Math in online casinos

Let us do some math to work out our chances in casino games. Casinos do this as well, for reasons explained later on in the article. Top online casinos do the same but for them it is for slightly different reasons. The best online casinos need to know all this math to incorporate it into the software, for every game from online poker to roulette.

With standard deviations (SD) you can quantify the luck factor in a casino game. With binomial distribution you can calculate the standard deviation of a simple game like roulette. Standard deviation is calculated, at least in binomial distribution, using the following formula: SD = sqrt (n*p*q), n meaning number of rounds, p meaning probability of winning and q meaning probability of losing. In binomial distribution, you assume that a result of 1 unit is a win and 0 units is a loss, instead of -1 units. This meaning the range of possible outcomes is doubled. Also we have to take into account that if we bet with 10 units the possible outcome increases tenfold. So,

SD (Roulette, betting on even numbers) :: 2b sqrt(npq ), b meaning the flat bet per round, n meaning the number of rounds, p equals 18/38 and q equals 20/38.

Example given, say after 10 rounds betting \$1 per round, The formula will be thus :: 2 x 1 x sqrt(10 x 18/38 x 20/38) = \$3.16.( b equals 1 and n equals 10 the rest does not change.) Also when 10 rounds have passed, the expected loss can be calculated using, n*b*h (n meaning rounds, b meaning the betting amount, h meaning house edge), so 10 x \$1 x 5.26%(house’s edge in roulette) = \$0.53. From this example we can deduce that the standard deviation is many times the magnitude to that of the expected loss.

That value is roughly almost six times the SD: three above the mean, and three below. So using above example still, 10 rounds later and we have betted \$1 per round, the result we would get is going to be somewhere between ((-h - (1/2(SD/h)) * SD) and (-h + (1/2(SD/h)) * SD)): -\$0.53 - 3 x \$3.16 and -\$0.53 + 3 x \$3.16, answer is between -\$10.01 and \$8.95. Of course you can lose more and win more than this, the chance for both is 0,1% so do not count on it. So after all of this horrible mathematics luck is quantified, so we know that when we bet \$5 all night long we are not walking out with \$500 at the end of the night.

You can calculate the SD for even more games, and the results are that Pai Gow has the lowest and Slots has the highest.  This is due to that when the potential payout increases the standard deviation increases as well.

If you increase the number of rounds drastically the expected loss will exceed the standard deviation many times over. This is not hard to deduce if you know how to look, but the standard deviation is dependent on the square root of number of rounds and the expected loss is only dependent on the number of rounds. This means that if the number of rounds increase to let us say near infinite then the expected loss would raise much more faster rate. So we can conclude mathematically that it is impossible to win for a gambler on the long term. Because the high ratio of SD to expected loss on the short term the gamblers are fooled into thinking they can win.

The important thing for a casino is to know the variance and house edge for every game. The percentage of the turnover that will result in their profit is equal to the house edge. The cash reserves needed in the casino that is equal to the variance. All the mathematicians and programmers who are calculating these things and making the formulas are called gaming mathematicians and gaming analysts. Casinos do not have these specialists in-house so they need to outsource this work.