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Gambler's Illusion in Roulette: Red or Black?

Let us bring about a frequent illusion of Roulette players. It consists in their expectation that if the same color has come up several past spins then it is more likely for the other color to appear. Unfortunately this is just a self-delusion that can lead to a tough landing.

This illusion is also promoted by casinos. Why do you think the casinos show the statistics of the past spins? Do you think that such statistics has a practical value, not to say can be used as a guide for prediction of the upcoming numbers and thus their colors. For novices to Roulette: each number has a fixed color that does not change and croupier announces the winning number and its color, for instance "Seven, red".

If for example a black number (any of the black numbers) came up ten times in a row, do you think it would raise your winning chances if you bet on the red color? Unfortunately not. This time is worthwhile to familiarize with two terms: a unique event and a series of events.

Unique Event

An event (a standalone / individual / unique / new event) is a designation for a random trial such as e.g. a spin in Roulette. If we forget about a zero (which is marked green) then the outcome of this event may be either red or black color. Each Roulette spin is a new or unique event. And this is the point of the whole illusion or one might say delusion. Every unique event starts over again regardless of the past. There is a saying that might be very handy for you to remember: "Roulette has no memory."

As every spin always presents a brand new event, red or black color can come up with the same probability at any time during the course of play, regardless of the fact that one of these colors has come up e.g. 20 times in a row. More precisely, if we play French Roulette with 18 red + 18 black + a single zero (there is an extra (double) zero in American Roulette, which only worsens winning chances), then the probability to win a bet on a color is always 18/37 = 0.4865 = 48.65%.

Both colors can come up with still-the-same probability. As it may seem like a paradox and as it can be expected "in reality" that the color will change at last, from a probability point of view it is completely groundless to assume that the color, which has not come up for a long time, is now more likely to appear!

Roulette feels no incentive to change the color. Although it holds true that if we spun the wheel numerous times (or close to the infinity), then the red should come up at approx. 48.65% cases, as well as the black color, and the rest of the cases would belong to the zero.

Additionally the assumption that the color must change may be very tricky. We tested the famous Roulette system called Martingale on our website. Its principle is simple: we bet on a color and double our previous bet in case of a loss until our color comes up. We always bet on the red color (by now we should know it does not matter, which color we pick or whether we alternate them or not). Black color and a zero were losing. Take a look below at the longest losing series. If you do not feel like counting, it took 19 loses (lost spins) in a row. Furthermore you can have a look at the longest Roulette series that were reliably recorded in stone casinos worldwide.

Would you imagine how much would such negative series cost, if you insisted stubbornly that the red color had to come up already? If you made an initial $5 bet, which is a minimum even-money bet at some casinos, and "20, black" appeared on the Roulette wheel, in the second spin you would have to bet $10.

Can you make a guess of how far would it go? At the end of the series (a zero was the 19th number) you would have to bet (and lose) $1,310,720 and altogether it would cost you over $2.6 million! This illusion, that you can win easily using this strategy, would, for certain, lead to a tough disillusionment. However the situation would not go that far likely as casinos set limits for the maximum bets.

Series of Events

The unique event described in the previous chapter must be distinguished from a series of consecutive events. As for the series it indicates that we consider what happened before. The probability of the series of events is determined by a multiplication of individual (unique) probabilities.

For example: what is the probability that the red color comes up three times (or in general x-times) in a row? Let us first clarify the unique probabilities in the spirit of the previous chapter. The probability for the red color to come up in the first spin is 18/37, and it is the same in the second and the third spin as well, because they always represent a unique event.

However the question is what is the probability of the red color to appear in three consecutive spins (that is now before the first spin is done). And that indicates the series of consequent events. It starts before the spin no. 1 and ends after the spin no. 3. The probability of the series of events is determined by the multiplication of the unique (or individual) probabilities, so the probability for the red color to come up three times in a row is 18/37 ˟ 18/37 ˟ 18/37 = (18/37)³ = 0.1151 = 11.51%. And that is the difference between a unique event and series of (consequent) events.

In the above-mentioned test of the Martingale system there was recorded the longest winning series too. Red color came up in 15 consecutive spins – see the figure below.

Once again, if we asked the question of what is the probability for the red color to appear 15 times in a row before the first number of this series (19) was spun, then we would calculate it as follows: (18/37)¹⁵ = 0.000020233. This low probability could be put in this way as well: 1 in 49,424. The figure 49,424 (to one) presents the fair odds that should be given by a betting company in case this hypothetical bet existed.

There is no such direct bet on the series in Roulette. However you can bet indirectly using the Anti-Martingale system. It consists in the fact that you pick up a length of series in advance, e.g. five black numbers, and you leave the initial bet including wins on the black color until the series is completed or broken (lost). If you succeed to finish the series, the win will grow exponentially. If the series is not competed at any time, you lose the initial bet only.

Conclusion

The idea that a change of colors must arrive is merely illusive and might end up with hard landing. The article aims at sending this important message: to realize and to remember that each Roulette spin presents a brand new event and that chances for both red and black colors to come up are entirely equal at any time, regardless of the fact that one of the colors have come up in the several past spins.

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