Let´s look at a European Roulette wheel. Over a single spin, the odds of a red number coming up are 18/37, or 1 in 2.06. Over 2 spins, the odds of 2 reds coming in are 1 in 2.06×2.06, or 4.24. The odds of seeing 10 reds in succession are 1 in (2.06) to the power of 10 = 1 in 1376. The odds of hitting 9 reds in a row on the next 9 spins on an American style (double zero) roulette. Wheel is about 0.12%. The odds of hitting 9 red in a row at some time depends upon how long you stay at the wheel.

By Ion Saliu, Founder of Roulette Mathematics

1. Without doing math to comment on the accuracy or inaccuracy of the '654,000 to 1' that you have posted for 18 reds in a row, the reason why waiting for 12 reds does not improve your odds is that it is the the point from which you are betting that the odds are '654,000 to 1' that you will hit ANOTHER 18 reds in a row. The odds of hitting red on.
2. On each spin of the ball, the chance of red coming up is 48.7%. On a single zero wheel, if a bet on red wins.487% of the time, the odds of two consecutive red spins is.237, three in a row is.115, four in a row is.0562, and five in a row is.0274. That means a streak of five happens about once in 36.5 spins, and we’ll round up to 37.

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'.

'Obviously in roulette betting on in the long run you are going to lose your money but at some point chances are you'll be in profit. To take an extreme example if you had $1000 you could reasonably expect to be up $1 at some point. Is it possible to generalize this? I want to win W dollars at which point I will quit. How much cash C would I need to have probability P of succeeding? Let's say I'm betting on a 37 number roulette wheel (18 red 18 black and one green 0)?'

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!

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!

Axiomatic one, everybody knows that the casinos have an edge or house advantage (HA) in all the games they offer, roulette including. The house advantage is created by the payouts in rapport to total possibilities for the respective bet. We can apply this simple formula based on units paid UP over total possibilities TP:

HA = 1 – (UP / TP)
(always expressed as a percentage.)

For example, in single-zero roulette, the one-number (straight-up) bet has payout of 35 to 1. The to qualifier is very important: the casino pays you 35 units and they give you back the unit you bet; thus, you get 36 units. There are 37 possibilities in single-zero roulette: 36 numbers from 1 to 36 plus the 0 number. Therefore, HA = 1 – (UP / TP) = 1 – (36 / 37) = 1 – 0.973 = 0.027 = 2.7%.

Let's calculate HA for the 1 to 1 bets: black/red, even/odd, low/high. HA = 1 – (UP / TP) = 1 – (2 / 2.055) = 1 – 0.973 = 0.027 = 2.7%. There are little differences among bets depending on how many decimal points we work with in our calculations.

The point is, the casinos have an advantage, or the players have a disadvantage. Nonetheless, the players' disadvantage is far better than what they face in state-run lotteries. Yet, most casino gamblers lose big, including at roulette tables. They do not have sufficient bankrolls to withstand long losing streaks.

However, around 45% of the roulette numbers lead the gamblers to profits in a few thousand spins. That is, with a sufficient bankroll, a player has a pretty good chance to make a profit, even if playing a random roulette number, or a favorite number. I analyzed about 8000 roulette spins from Hamburg Spielbank (casino). Quite a few numbers ended up making a profit: roulette systems, magic numbers.

By contrast, the more lottery drawings a player plays, the higher the degree of certainty of a loss. Let's make a comparative analysis to the roulette long series above (spins: total roulette numbers, 37, multiplied by 200). If playing the pick-3 lottery for some 100,000 drawings, it is guaranteed that all pick-3 straight sets will be losers. Some numbers will hit up to 3% to 5% above the norm — but that is not nearly enough to assure a profit. A frequency of 3% to 5% above the norm leads to profits in roulette, however.

Ion Saliu's Paradox and Roulette

Ion Saliu's Paradox of N Trials is presented in detail at saliu.com, 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.

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!):

www.spielbank-hamburg.de/spielsaal/permanenzen.php4

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

www.spielbank-wiesbaden.de/DE/621/Permanenzen2.php: Wiesbaden Spielbank Permanenzen

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Here it is... from your 'parent' site....an answer given to a question....
What would be the odds of having a red number turn up 18 times in a row on Roulette. - Doug Jeffrey from Eugene, USA
(18/38)18 =~ 1 in 693745. Nov. 11, 2001
how can you apply a forward prediction like that???... but then not be theoretically allowed to intercept at the 12 in a row point and bet against a 1 in 693745 chance?? Please explain

If you 'jump in' after 12 red spins in a row, the probability of there being a streak of 18 reds (where the last 12 spins are the first 12 of the streak) is much higher, calculated as follows:

Odds Of 7 Reds In A Row Roulette Rules

(1/1)^12 * (18/38)^6 = 0.011296 = 1 in 88.525125
The original question was referring to 18 future spins. If 12 have already occurred, then it is only a question of 6 future spins.
The first 12 were red, so the probability of those spins being red is 100%, not the 18/38 per spin that it was before their outcomes became known. The probability of the 6 unknown future spins being red is still 18/38 per spin.

and...if you look at this situation after it has happened... 18 spins..18 reds...are you then allowed to look back and say...what happened there was a one in half million...or whatever that figure is...of happening?????

You could look back at the last 18 reds and say 'That particular sequence had a 1 in 693,745 chance of occurring' ... but you can also say the exact same thing about any 18-spin sequence where no numbers were green.
Just keep in mind that it has no bearing whatsoever on what will happen on future spins. The past does not matter.

JB...thanks for your help on this...I'm keen on a reply on the last point....what yu said in your last reply....unfortunately has me believing....that you can legitimately expect to win....if you jump in at 12 reds...and bet the next 6 blacks....based on a theory that 18 reds in a row is a one in approx half million chance!........and on those 6 bets you stake it small to win small....over and over......in 20 000 spins...you'll risk $50.....to win $30.......might you win significantly more often than you lose?

Regardless of when you 'jump in', your odds for the next spin are always the same:
red: 18/38
black: 18/38
zero: 2/38
In the long run, you will always lose 5.26% of whatever you bet, regardless of when you bet. (Assuming double-zero roulette)

I see that....Let me ask you this question JB....does it confuse you at all that you can look back on a sequence and accurately say those 18 spins in a row being even was a 1 in half million chance.......but .......that said......you cannot jump in at 12 evens in row and bet against that sequence of 18 happening.....does that confuse you just a little...or is it clear to you....it might be a state of mind thing that I just cannot grasp....

It doesn't confuse me, because the past does not affect the future when you're talking about random, independent events such as spins of a roulette wheel.Let's attack this from another direction.

Odds Of 7 Reds In A Row Roulette Games

Assume that somewhere there is a casino that has a jackpot bet for 18 Red/Black/Odd/Even/High/Low in a row.
Do you think that the casino would let you place your bet after 12 in a row had already hit?

Odds Of 7 Reds In A Row Roulette Numbers

Remember: For every gambler that looked at the display at the roulette table and said 'Look honey. Red came out 12 times in a row! We should bet on black,' there was another sucker that said 'Look honey. Red came out 11 times in a row! We should bet on black.'