Roulette math


The most important parameters to consider when analyzing a game are:


**HA (House Advantage)**


This value is a function of the payoff (x to 1) and the actual odds (y to 1). It is calculated very simply:


(y-x)/(y+1). It is a percentage value.


Any bet in roulette has a negative expectation. If you win the round, you make a profit, but this profit is less than it would have been in the case of a fair game.


**EV (Expected Value)**


This value is a function of the probability of winning and losing, and the net payoff. It is calculated as ∑ (net payi x pi), where payi is the net payoff and pi is the probability of the payoff. It is a numerical value.


**SD (Standard Deviation)**


The parameters mentioned above are theoretical; in practice, an extremely important factor must be considered: volatility.


The analysis of volatility is an important tool for both managers and professional players. Both categories should be able to understand what constitutes a normal deviation and what might raise doubts, leading to further investigations.


To understand what fluctuations can be considered normal, a good starting point is a common statistical measure called standard deviation. The standard deviation indicates how much variation can be expected when observing repeated occurrences of a random variable.


If there were no absorbent barrier (maximum bet), the Martingale strategy would be winning despite the negative EV of each bet because it only requires one win to end the game. The probability of not winning a bet tends to zero as the number of rounds played increases.


The house has an advantage over the player higher than in other casino games (baccarat, blackjack).


However, roulette, especially the French version, is much slower than blackjack and mini baccarat. With the same average bet, the hourly EV (Expected Value) of roulette does not necessarily turn out to be more disadvantageous than other games.



**RED-BLACK EVEN-ODD HIGH-LOW (to split in case of 0)**

Winning probability: 48.6%, split probability: 2.7%

Payout: 1 to 1

House advantage: 1.35%

Standard deviation: 0.99



Winning probability: 32.43%

Payout: 2 to 1

House advantage: 2.70%

Standard deviation: 1.40



Winning probability: 16.22%

Payout: 5 to 1

House advantage: 2.70%

Standard deviation: 2.21



Winning probability: 10.81%

Payout: 9 to 1

House advantage: 2.70%

Standard deviation: 2.79



Winning probability: 8.11%

Payout: 11 to 1

House advantage: 2.70%

Standard deviation: 3.28



Winning probability: 5.41%

Payout: 17 to 1

House advantage: 2.70%

Standard deviation: 4.07



Winning probability: 2.70%

Payout: 35 to 1

House advantage: 2.70%

Standard deviation: 5.84


For American roulette, the house advantage is:




**MULTIPLE CHANCES** (excluding the first 5): 5.26%


**THE FIRST 5 (0 00 1 2 3)**: 7.89%


By simple chances, we mean RED/BLACK, EVEN/ODD, HIGH/LOW, and in case of 0 or 00, the player loses the entire bet.