For the calculation of Roulette odds, a little skill at mathematics is useful, but it is certainly not rocket science. Just a few basic equations can reveal everything a player might wish to know about possible outcomes, from series of simple straight-up bets to more complex wagering patterns. In the following example, it assumed that the European form of the game with 37 numbers is being played, whereas it offers a lower House advantage than the 38-number American version.

As Easy as P = B/37

The probability (P) of a winner coming up on a single spin of the wheel is simply the sum of all numbers bet (B) divided by the total number of slots on the wheel (37). It can be expressed either as a ratio, B:37, or by the formula P = B/37. To calculate the odds of a single specific number coming up (B=1), the ratio would be 1:37, and the equation would yield P = 1/37 = 2.70%.

The same approach can be used to calculate the odds of one of the 18 red numbers coming up (B=18). The ratio would be 18:37, and the equation would show that P = 18/37 = 48.65%. This calculation also applies to all of the other even money bets: black, odd, even, low (1~18), and high (19~36). The player can expect to win a bit less than half the time, with the difference being the House edge.

Using this simple formula, there should be no trouble calculating the probabilities of a winning split (pair) must be 5.4%, a winning street (trio or row) will be 8.1%, and a winning column of 12 numbers or a sequential dozen is 32.43%. Expressed as ratios, the odds are 2:37, 3:37 and 12:37, respectively.

Multiple Spin Calculations

Having mastered the equation for calculating the probability of an outcome for one spin, it is now possible to apply the math to multiple spins to see if a winner is likely to come up within a specific series of bets. In this regard, it is important to remember that Roulette is a game of independent events. The game has no memory, which means past results have no influence on future results.

As an example, consider the popular betting pattern called the “Birthday Strategy,” where the player wagers on the numbers corresponding to the day and month of his/her birth. A person both on February 26^th would bet on 2 and 26, while someone born on November 3^rd would bet on 3 and 11. In both cases, B=2, so the odds of winning on a single spin are P = 2/37 = 5.4%.

However, the Birthday player is committed to wagering on this outcome more than once. For every spin the two numbers are bet on, the probability of one coming up will always be the same, namely 5.4%. It is therefore possible to calculate the odds of a winner for any numbers of spins (N) by using second, slightly more complex equation: P = 1 – {(37-B)/37}^N.

Here’s how it works. If a player is willing to risk a chip on each of the two numbers (B=2) for 13 spins (N=13), that’s 26 chips in total required. The equation can be used to show that the likelihood of success is P = 1 – {(37-2)/37}¹³ = 51.4%, or a little more than half. How much is won on this wager depends on how soon the winner comes up, but it will pay no less than 35-to-1 for the winner minus the total previously wagered and lost, which can be no more than 25 chips.

In fact, the equation can be used to show that this strategy will be successful anytime one of the two numbers comes up within 17 spins, since it requires the risk of no more than 34 chips and will pay back 35-to-1. In this case, P  = 1 – {(37-2)/37}¹⁷ = 61.1%, which implies that a “birthday present” can be expected a little less than two-thirds of the time this simple strategy is played.