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Using Expected Value for Roulette Outcomes

Expected value is a mathematical formula which can be used to estimate what players can expect to reap from a gambling game. This theory of expected value can be used to analyse the possible outcomes of roulette. The concept uses the probability of each out come and the number of outcomes to calculate what players can expect to gain or lose in a roulette game. The probability of an accurate bet is 1 out of 37 or 38 depending on what kind of roulette game you are playing.

Using the formula of expected value, we can easily calculate how much you can stand to lose when betting on roulette. This is done suing probabilities of the ball landing in to each sac. We will examine each expected value of the different online roulette variants using examples.

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How to Use Expected Value Statics for Roulette

The game of roulette whether online is a random game of chance. However as with most things with numbers you can calculate a pattern of sorts for lack of a better word. There are 37 to 38 possible outcomes in a roulette. So, for each spin of the wheel a player has once chance out the total number of ball pockets to win. Then narrow it down to colours then the player has one chance out of the total number of ball pockets of their chosen colour. Inversely there are more pockets of the opposite colour and the zero pocket in which the ball has a chance of landing. All these probabilities can be used to determine how much money, in the event of continuous player, players will lose when playing roulette.

Calculating Expected Value for American Roulette

American Roulette wheels have 38 equal size pockets where the ball will eventually land after the wheel is spun. There are three main colours on the wheels, green, red and black. The two green spaces have numbers 0 and 00 respectively on them. The other 36 spaces are numbered from 1 to 36. The 36 are split evenly between red and black. Players can wager on either the ball landing on a colour or a number. The latter is more difficult but has a higher reward.

The ball has an equal likelihood to land in any of the pockets since the spaces are the same size, the roulette wheel outcomes have uniformly distributed probability. In the event that a player has bet $1 on the ball landing in a red pocket, the expected value will be calculated with the following probabilities


An American roulette wheel has total of 38 spaces. The probability that a ball lands on one space is 1/38.

There are 18 red spaces, and the probability that the ball lands on red is 18/38.

There are 20 black or green pockets (18 black and two green), and so the probability that it does not lad on a red is 20/38.

If the ball lands on red the player wins in total $2; their original dollar back and net winnings of $1. If the ball lands in green or black, then the dollar bet is lost resulting in net winnings of $-1.

Calculation of Expected Value

We use the above information with the formula for expected value. Since we have a discrete random variable X for net winnings, the expected value of betting $1 on red in roulette is:


Probability of Red x (Bet amount) + Probability of green or black (Not Red) x (Value of lost bet)

18/38                    x              $1           +                             20/38                                    x              ($-1) = -0.05263

The expected value is -0.05263 which is just over a negative 5 cents. Therefore, in essence when you play roulette over a period you must expect in theory to lose 5.3 cents per dollar for every bet on red.

The expected value changes as the variables change. For instance, when you bet on a number instead of a colour in American Roulette the equation changes. Using the same formula:

Probability of your number x (Bet amount) + Probability of another number x (Value of lost bet)

1/38                                       x              $1           +             37/38                    x                              $-1= $-0.947

When you bet on a number the expected los per dollar is almost doubled to $-0.95 / 95 cents


Expected Value for French and European Roulette

When calculating expected value for French and European roulette you use the same formula. Just keep in mind that the wheels for French Roulette and the European variant have only 37 pockets in total. So, using our previous variables from the American Roulette examples here’s a demonstration of expected value in French and European Roulette.

Bet on Red

Probability of Red x (Bet amount) + Probability of green or black (Not Red) x (Value of lost bet)

18/37                    x              $1           +                             19/37                                    x              ($-1) = $-0.027

Bet on a number

Probability of your number x (Bet amount) + Probability of another number x (Value of lost bet)

1/37                                       x              $1           +             36/37                     x                              $-1= $-0.945

In essence expected value is a calculation of house edge. This is not to say every time you play Roulette you will lose that exact amount. In some instances, you may win multiple times in a row. The expected value is just an average of what you will lose in prolonged continuous play. The amount you do lose is insignificant however at the end if theoretically you were to play the house would have gained more. That is where the game of roulette is exciting because you could get really lucky. Try out your luck with any of the three types of the game at the best NZ casinos we recommend.




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