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Probability, Odds to Win at Roulette in Various Number of Spins ~
To Be Ahead of House Edge and Quit the Roulette Table as a Winner

By Ion Saliu, Founder of Roulette Mathematics

Winning roulette: Probability, odds of number of spins, streaks to leave roulette tables a winner.

This question was posted in mathematical newsgroups (alt.math.recreational, alt.math.undergrad, alt.sci.math.probability): "Winning and Quitting on Red/Black in Roulette".

On the surface, the best probability for the roulette player to be ahead is in one trial (spin): 48.6% to win (versus 51.4% to lose), as far as even-money betting is concerned. I don't agree that it is the best strategy (betting all your bankroll on one spin).

Theoretically, no bankroll will put a player ahead guaranteed, IF flat-betting and playing very long consecutive sessions. There are moments, however, when the roulette player can be ahead by at least one betting unit. Even in even-money bets, the player has a good chance to be ahead by at least one unit after 5, or 10, or even 100 spins. But more than 20 spins are NOT recommended; the probability (odds) to lose go(es) above 50%! Think about it!

The main thing, mathematically, is the number of player's wins in N trials. To be ahead, means the player has won at least one more roulette spin (number of successes) than the number of losses in N trials. The question then becomes:

"What are the probabilities for the player to be ahead in various numbers of trials?"

Everybody can use my probability software SuperFormula, option L: At Least M successes in N trials.
Winning probability: p = 18/37; M must be at least (N/2) + 1.
Here is a number of cases from the player's perspective.

The figures are applicable to all even money roulette bets: black or red; even or odd; low or high (1-18 or 19-36).

1 trial (spin)
- probability (odds) to win: 48.6%; odds = 1 in 2.05
- probability (odds) to lose: 51.4%; odds = 1 in 1.95
(the probability to lose is 19/37; adding zero to unfavorable cases).

2 trials (spins)
- probability (odds) to win 2 of 2: 23.7% (1 of 2 doesn't mean 'being ahead')
- probability (odds) to lose 1 of 2: 76.3%

3 spins
- probability (odds) to win at least 2 of 3: 48%
- probability (odds) to lose at least 2 of 3: 52%

10 spins
- probability (odds) to win at least 6 of 10: 34.4%; odds = 1 in 2.91
- probability (odds) to lose at least 6 of 10: 41.1%; odds = 1 in 2.43

20 spins
- probability (odds) to win at least 11 of 20: 36.5%
- probability (odds) to lose at least 11 of 20: 46.2%

100 spins
- probability (odds) to win at least 51 of 100: 35.5%; odds = 1 in 2.82
- probability (odds) to lose at least 51 of 100: 56.8%; odds = 1 in 1.76.
It's getting worse for the player...

The roulette strategy (or system) is a totally different ball game! But there are professional gamblers out there, including roulette players! They must have strategies, some roulette systems deduced from some figures like the ones above! The player can be ahead at any point in the game. If so, maybe it's time to move to another (or casino) table: It improves the odds of winning!

Mathematics of roulette translates to winning and losing streaks. Always keep track of the losing and winning streaks. Be strong and put an end to a winning streak. You are ahead, you quit the roulette table. Go to another table and wait until you are ahead. The bankroll is of the essence: It must assure going through long losing streaks. Divide the streaks in 10 spins or 20 spins. Never fight aggressively short or mid-term losing streaks. This is the best approach for those who do not know Ion Saliu's casino gambling systems. A good approach to gambling is the next best thing to a good gambling system! Applicable to blackjack and baccarat, too!

      Various odds (probabilities) at 0 and 00 roulette

 Type of Bet                Single 0     Double 0

 - straight-up (1 number):  1 in 37      1 in 38    
 - split (2 numbers):       1 in 18.5    1 in 19    
 - street (3 numbers):      1 in 12.3    1 in 12.7  
 - corner (4 numbers):      1 in 9.25    1 in 9.5   
 - fiveline (5 numbers):    1 in 7.4     1 in 7.6   
 - sixline (6 numbers):     1 in 6.2     1 in 6.3   
 - column (12 numbers):     1 in 3.1     1 in 3.2   
 - dozen (12 numbers):      1 in 3.1     1 in 3.2   
 - black/red (18 numbers):  1 in 2.06    1 in 2.11  
 - even/odd (18 numbers):   1 in 2.06    1 in 2.11  
 - low/high (18 numbers):   1 in 2.06    1 in 2.11  

Ion Saliu's Paradox and Roulette

Ion Saliu's Paradox of N Trials is presented in detail at, especially the probability theory page and the mathematics of gambling formula. If p = 1 / N, we can discover an interesting relation between the degree of certainty DC and the number of trials N. The degree of certainty has a limit, when N tends to infinity. That limit is 1 1/e, or approximately 0.632....

If you play 1 roulette number for the next 38 spins, common belief was that you expected to win once. Not! Non! Only if you play 38 numbers in 1 spin, your chance to hit the winning number is 100%. Here is an interesting table, which includes also The Free Roulette System #1 presented at the main roulette site.

Shop best software books: Lottery, Powerball, Mega Millions, gambling blackjack, roulette, sports.

The maximum gain comes when playing 38 numbers in one spin: 36.3%. Obviously, it makes no sense to play that way because of the house advantage. On the other hand, a so-called wise gambler is more than happy to play one number at a time. What he does is simply losing slowly! Not only that, but losing slowly is accompanied by losing more. That cautious type of gambling is like a placebo. A roulette system such as Free System #1 scares most gamblers. "Play 34 or 33 numbers in one shot? I'll have a heart attack!" In reality, the Free Roulette System #1 offers a 28.8% advantage over playing singular numbers in long sessions. That's mathematics, and there is no heart to worry about, axiomatic one.

You can also use SuperFormula to calculate all kinds of probabilities and advantage percentages. The option L At least M successes in N trials is a very useful gambling instrument. If you play 19 numbers in one spin, the probability to win is 50%. If you play 19 numbers in 2 consecutive spins, the probability to win at least once is 75%.

Editor's note
In an apparent change of heart, the Hamburg casino (Spielbank) offers online roulette results for free. The new link is (for the time being!):

Real-life roulette spins are also available from the Wiesbaden, Germany, Casino (Spielbank) Wiesbaden Spielbank Permanenzen

Study the advantages of Ion Saliu's Paradox in the game of roulette.

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