This article says about how to calculate mathematical expectations (casino’s advantage) for different stakes at Roulette and what determines this advantage. Let’s take for example the American Roulette i.e. we have a wheel with 38 numbers and 11 different variants of stakes. As the theory of probabilities tells us, we deal with independent events i.e. the outcome of each spin is not connected with the previous result and doesn’t influence on the next one. It means that if the black has dropped ten times in succession, chances to get a red by the next spin are equal to the probability of getting a black. The probability of getting a definite number is 1 out of 38 or 37 by 1.

Now be attentive: the formula of mathematical expectations is [the number of favorable outcomes/the number of possible outcomes] X Payment by the favorable outcome + [the number of unfavorable outcomes/the number of possible outcomes] X stake.

Then compare, you make a bet of $1:

Straight up = (1/38) x 35 + (37/38) x (-1) = -2/38 = -0.0526 or (-5.26%);
Five numbers = (5/38) x 6 + (33/38) x (-1) = -3/38 = -0.0789 or (-7.89%);
Red bet =  (18/38) x 1 + (20/38) x (-1) = -2/38 = -0.0526 or (-5.26%);

The negative result can be explained by that we have calculated the mathematical expectation for a player who makes a bet. Change the bar and get what is called the casino’s advantage.