How many times in a row has a little ball landed in the same pocket of a Roulette wheel, i.e. how many times has a single number occurred in a row? And how about the same color? What is the probability of these events and a potential impact on a play?

The probability that any single number occurs is 1/37 in French Roulette and 1/38 in American Roulette (there are 36 numbers + zero + double zero in American Roulette). There is no doubt that it is a great coincidence when the same number comes up again and again.

**The longest reliable series** was registered at the hotel El San Chuan in Puerto Rico on 9 June 1959. During the course of the American Roulette, number ten occurred even **six times in a row!** The probability of such (successive) events is determined by a multiplication of individual events. Therefore the probability that the same number comes up six times in a row is:

`(1/38) ˟ (1/38) ˟ (1/38) ˟ (1/38) ˟ (1/38) ˟ (1/38)`

, that is:

`(1/38)`

.^{6} = 0.000000000332122593261671

That is a very small number indeed, roughly three billionths only. If we convert this probability into true odds that would have to be offered to us by a casino, we get the value `3,010,936,384 to one`

. The true (fair) odds are calculated as a reciprocal of the probability, that is 1 ÷ probability. If such a bet on a series of outcomes was possible in Roulette, we would win $3 billion for a $1 bet(!)

It is important to add that the above-mentioned calculation of probability deals with a *multiple (successive) events*, i.e. we can ask this question: What is the probability that the same number in Roulette comes up 6 times in a row?

Since it would be a different case if e.g. number 10 occurred and after that before the new spin we asked what was the probability that number ten came up again? In this case the answer would be 1/38 (in terms of American Roulette), because any number could occur with the same probability 1/38 in every new spin. That is what we call a *simple event* in contrast with a multiple event(s) whereas the probabilities of individual events are multiplied (→ Articles on Probability).

The true odds for a 1 to 10fold repetition of the same number are shown in the table below. It is the same mechanism as if a sporting bet company or a casino offered the odds for a victory of some home team in some football match (→ The Odds Determination and Calculation).

The Same Number Comes Up in a Row | True Odds to One in FRENCH Roulette | True Odds to One in AMERICAN Roulette |
---|---|---|

1˟ | 37 | 38 |

2˟ | 1,369 | 1,444 |

3˟ | 50,653 | 54,872 |

4˟ | 1,874,161 | 2,085,136 |

5˟ | 69,343,957 | 79,235,168 |

6˟ | 2,565,726,409 | 3,010,936,384 |

7˟ | 94,931,877,133 | 114,415,582,592 |

8˟ | 3,512,479,453,921 | 4,347,792,138,496 |

9˟ | 129,961,739,795,077 | 165,216,101,262,848 |

10˟ | 4,808,584,372,417,850 | 6,278,211,847,988,230 |

The odds are reciprocal values of the probabilities – the higher they are, the lower the probabilities are. The case of the above-mentioned record series is marked green. Consider also **the difference that is made by one extra number** in American Roulette (the double zero).

There are no exceptions that the same color appeared more than 20 times in a row in practice. The record was registered in 1943, when **red color came up 32 times in a row!** The probability of such event in French Roulette is `(18/37)`

with the corresponding odds ^{32} = 0.000000000096886885`10,321,314,387:1`

.

The probability of the 32fold repetition of the same color in American Roulette is much more lower: `(18/38)`

and the odds are ^{32} = 0.00000000004127100756`24,230,084,485:1`

. Thus this is even less likely than occurrence of a single number six times in a row. Again it is clearly demonstrated what kind of importance (a negative one for players) has just one extra number in American Roulette.

Now imagine that you used the Martingale betting strategy (→ see the first test of the Martingale system), whereas the next bet is doubled if your bet loses...

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