Roulette - General Questions

In roulette any bet or combination of bets carries a high house edge. The more likely you are to win the more you will have to risk relative to the reward. If you do this 10 times the probability of showing a profit is 46.42%. At 100 times the probability drops to 24.6%.

Start the same way as I explained with even money bets. Each bet should be the sum of the left and right numbers. However, following your two column strategy you should add double the amount to the right if you lose.

I played it in practice mode and it seems to be a legitimate no-zero roulette wheel. There is no system that can either beat or lose to this game in the long run. The more you play the more the ratio of the net win to the total amount bet will get closer to zero.

Update: This casino has since closed.

If you double on 11 at a Microgaming casino and the dealer gets a blackjack you will lose the total amount bet. At Unified Gaming the blackjack would be turned over immediately if the up card were a ten so there is no risk of losing to a blackjack when doubling in this situation.

All combinations of bets in roulette yield the same expected return, assuming the dreaded five-number combination is avoided. You're right that the third column has eight reds and four blacks. The probability of winning 3 units is 8/38, 1 unit is 4/38, breaking even is 10/38, and losing 2 units is 16/38. The combined expected per unit bet is return is (1/2)*(3*8 + 1*4 + 0*10 + 2*16)/38 = -2/38. Betting on black and the third column the probability of winning 3 units is 4/38, 1 unit is 8/38, breaking even is 14/38, and losing 2 units is 12/38. The expected return is (1/2)*(3*4 + 1*8 + 0*14 + 2*12)/38 = -2/38. Both combinations weight the various outcomes differently, but they average to the same number.

Once you have hit n numbers, the probability of getting a new number on the next spin is (38-n)/38. If the probability of an event is p, then the expected number of trials before it happens is 1/p. Thus the expected number of spins to get a new number, given that you already have n, is 38/(38-n). For example once you have hit 20 numbers the expected number of spins to get the 21st is 38/18=2.11. So the answer is the sum of the expected number of spins at each step: (38/38)+(38/37)+(38/36)+...+(38/1)=160.66.

(18/38)¹⁸ =~ 1 in 693745.

I’m happy to have helped the main in the white coat. My roulette advice is limited to games on earth, bribery is recommended on your planet.

The probability of your number hitting exactly x times is combin(1000,x)*(1/38)^x*(37/38)^1000-x. The following table shows the probability of all number of hits from 0 to 6 and the total.

Wins in 1000 Roulette Bets

Number	Probability
0	0.00000000000262
1	0.00000000007078
2	0.00000000095556
3	0.00000000859146
4	0.00000005787627
5	0.00000031159330
6	0.00000139655555
Total	0.00000177564555

So the answer is 0.00000177564555, or 1 in 563175. I hope this didn’t happen at an Internet casino.

You may wonder why I didn’t use the normal approximation as I did with the coin flipping problem above. That is because it doesn’t work well with very high and very low probabilities.

You accessed my strategy quite well. I intended to bet the minimum through most of the hour to avoid the house edge grinding me down. The producer was not happy with such a boring strategy so I tried to act more excited than I normally would be. The reason I picked a single number as opposed to an even money bet is that I wanted volatility. I knew if I ended close to my starting point of $10,000 I would probably lose. So I wanted a bigger shot at getting ahead. I stuck with 23 the entire time, although towards the end I added 5 as well. I’m Glad you liked the show.

No. Dealers are taught only the basics and nothing nearly that skillful. In fact if a dealer had that control he could simply get an accomplice to bet wherever he planned for the ball to land and they could easily make millions.

The probability of losing any one spin is 1-(8/37) = 78.38%. So the probability of losing 15 spins is .7838¹⁵ = 2.59%.

First let me say how not to test any game. You should not record all your play and when you are done look for any kind of anomaly, and then write to me complaining that the casino is cheating. The correct way is to state a hypothesis for how the casino is cheating FIRST, then gather data, and finally see if the data fits your hypothesis. If you don’t know what kind of hypothesis to state I would suggest simply testing for the number of wins and losses, and then bet the same thing every time. If you do the first two steps properly and need help on the third then feel free to write me.

In your example the casino’s hold is 75%. That is the ratio of how the value of chips players walk away with to the amount of money the dealer drops in the box. Although there are historical averages there is no way to go from house edge to hold. The hold depends in part on how long the player will circulate through his chips, and there is no way to know that.

Thanks for the compliment. There is a formula for questions like this, which I explain on my site www.mathproblems.info, see problem 116.

With a bankroll of b units, winning goal of g units, probability of winning p, and probability of losing q your probability of success is ((q/p)^b-1)/((q/p)^b+g-1). In this case b=20, p=18/37, q=19/37, and g=20, 50, and 100. So for a bankroll of $200 the probability is ((19/18)²⁰-1)/((19/18)⁴⁰-1) = 0.253252.

For a winning goal of $500 the probability is ((19/18)²⁰-1)/((19/18)⁷⁰-1) = 0.045293.

For a winning goal of $1,000 the probability is ((19/18)²⁰-1)/((19/18)¹²⁰-1) = 0.002969.

It doesn’t make any difference what the past spins were. The probability of red 23 three times in a row is (1/38)³ = 1 in 1 in 54,872.

The probability of three pairs and six singletons in twelve spins is combin(38,3) × combin(35,6) × combin(12,2) × combin(10,2) × combin(8,2) × fact(6)/38¹² = 9.04%. The math gets rather messy asking about the probability that this could happen in any 12-spin span over 20 total spins. Suffice it to say that it is significantly more than 9%, more likely than not, I would guess. So these seem like very normal results to me.

The house edge on the specified number would be exactly 0%.

The probability of winning is (18/38)³ = 10.63%. The house edge is 8×0.1063 - 1×0.8937 = 4.34%, which is less than the 5.26% on all the other bets (except the dreaded 0,00,1,2,3 combination at 7.89%).

For the benefit of other readers, at many casinos in Europe, if you make an even money bet in roulette and the ball lands in zero, then the wager becomes imprisoned. If the next roll would cause the wager to win, then it is returned without winnings. If the ball lands the opposite way, then the bet loses.

What happens to single-imprisoned bets if the ball lands in zero? If the casino allows only single imprisonment, then it would lose. However, some casinos allow for double imprisonment, which is what would happen. If a double imprisoned bet wins, then it returns to being single imprisoned. If the casino allows only up to double imprisonment, then a double imprisoned bet would lose on another zero. By the same logic, if a casino allows triple imprisonment, then a double imprisoned bet would become triple imprisoned on another zero.

The house edge under triple imprisonment is 1.370120%. Before I explain how I arrived at that, let’s do single and double imprisonment first. Assume the first bet is on red.

Let Z = probability of a zero = 1/37.
Let R = probability of a red number = 18/37.

Single imprisonment

pr(push) = ZR = 0.0131482834.
pr(win) = R = 0.4864864865.
pr(loss) = 1-pr(push)-pr(win) = 0.5003652301.
Expected value = pr(win)-pr(loss) = -0.0138787436.

Double imprisonment

The player will push if the first spin is a zero, then the player can repeat ZR from 0 to an infinite number of times, and then get a red. In other words, the ways to push are:

ZR, Z(ZR)R, Z(ZR)(ZR)R, Z(ZR)(ZR)(ZR)R, ...

pr(push) = 0.013323464
pr(win) = 18/37 = 0.4864864865.
pr(loss) = 1-pr(push)-pr(win) = 0.5001900494.
Expected value = pr(win)-pr(loss) = -0.0137035629.

Triple imprisonment

First, let’s find the probability p₁ that a single imprisoned bet becomes triple imprisoned with two more zeros, and then eventually rises back up to being single imprisoned again. This can happen as follows:

ZZRR, ZZ(RZ)RR, ZZ(RZ)(RZ)RR, ZZ(RZ)(RZ)(RZ)RR, ...

In other words, the bet can bounce between triple and double imprisonment up to an infinite number of times.

Second, let p₂ = the probability that a single imprisoned bet reaches the first or second level of imprisonment and then returns to single imprisonment.

let p₂ = ZR + p₁ = 0.013323464.

The player can repeat returning to the first level from zero to an infinite number of times. So the probability of a push is:

ZR + Z p₂R + Z p₂ p₂R + Z p₂ p₂ p₂R + ... =

Z × (1/(1- p₂)) × R = 0.013325830.

pr(push) = 0.013325830.
pr(win) = 18/37 = 0.4864864865.
pr(loss) = 1-pr(push)-pr(win) = 0.5001876839.
Expected value = pr(win)-pr(loss) = -0.0137011974.

Infinite imprisonment

Not that you asked, Maff, but in Spain I hear they allow infinite imprisonment. Let p = probability of a push. This is also the probability of starting at level x of imprisonment, falling deeper in levels, but eventually climbing back up to x.

p = ZR + ZpR + ZppR + ZpppR + ...

p² - p + ZR = 0

By the Quadratic Formula p = (1-(1-4*RZ)^1/2)/2 = 0.0133258620.

pr(push) = 0.0133258620.
pr(win) = 18/37 = 0.4864864865.
pr(loss) = 1-pr(push)-pr(win) = 0.5001876515.
Expected value = pr(win)-pr(loss) = -0.0137011650.

Zeros ignored imprisonment

Finally, at some casinos zeros are simply ignored after the first zero which imprisons the bet. There the probability of a push is simply (1/37)×(1/2) = 0.0135135135.

The following table summarizes all four kinds of rules.

Imprisonment Odds

Imprisonment	Win	Push	Loss	Expected Value
Single	0.48648649	0.01314828	0.50036523	-0.01387874
Double	0.48648649	0.01332346	0.50019005	-0.01370356
Triple	0.48648649	0.01332583	0.50018768	-0.01370120
Infinite	0.48648649	0.01332586	0.50018765	-0.01370117
Zeros ignored	0.48648649	0.01351351	0.50000000	-0.01351351

I’d like to thank ChesterDog and weaselman for their mathematical help. I’d also like to give thanks and a plug to professor G. Artico and polarprof.it for the HTML of the summation formula above.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

The idea is that the wheel should be balanced. In other words, if you took the average of a section of consecutive numbers on the wheel, they would be close to the average number 18.5. I thought of a way to explain how the numbers are distributed on a double-zero wheel, as follows.

• Position the 00 at the 12:00 position and the 0 at the 6:00 position.
• The numbers at the top of the wheel are 13, 1, 00, 27, 10*, 25. This must simply be memorized.
• All low odd numbers (from 1 to 17) are on the left side and the high odds (19 to 35) on the right side.
• Starting with the 1 and 13, move four positions counter-clockwise to get the next odd, but do not pass the zero.
• As an exception to rule 4, do not place the 19 four positions counter-clockwise from the 17, because the 19 is a high odd and belongs on the right side. Instead, put the 11, the only low odd not otherwise covered in rule 4.
• All even numbers are directly across the wheel, 180 degrees, from the preceding odd number.

I'm at a loss to explain the order on a single-zero wheel. For now, I can say that if you put the zero in the 12:00 position, then:

• On the left side will be all red low numbers and black high numbers.
• On the right side will be all red high numbers and black low numbers.

This question is raised and discussed in my forum at Wizard of Vegas.

You would expect the number of times the ball would land in 1 to 12 to be 3000*(12/38) = 947.37.

The difference between your results and expectations is 947.37 - 742 = 205.37.

The variance is 3000*(12/38)*(1-(12/38)) = 648.20.

The standard deviation is the square root of the variance = sqrt(648.20) = 25.46.

Your results are 205.37/25.46 = 11.75 standard deviations south of expectations.

The p value, or the probability of being off by 11.75 standard deviations or more is 1 in 28,542,806,257,940,300,000,000,000,000,000.

I would be interested to know where the wheel is.

It's easy to see the purpose of that system is to cover most numbers so it's a fairly low risk way of playing roulette. Here is what to bet on each spin:

• $5 each on 3, 16, 24, 28, and 33.
• Make a corner bet on each of these sets of numbers: 2/3/5/6, 7/8/10/11, 14/15/17/18, 19/20/22/23, 26/27/29/30, 31/32/34/35.

Note that this does not cover the following nine numbers: 0, 00, 4, 9, 12, 13, 21, 25, and 36.

The following return table shows the probability and contribution to the return of all possible outcomes.

The lower right cell shows an expected loss of $9.21 per spin. The total amount bet per spin is $175. This results in a house edge of $9.21/$175 = 5.26%, the house edge in double-zero roulette.

I would like to add that you will probably lose more than you get back in comps with this strategy, or any roulette strategy. The rule of thumb is casinos will give you back about 1/3 of your expected loss in comps. There are ways of fooling the casinos that your expected loss is more than it really is, but playing this strategy isn't one of them.

I agree that the conditional advantage of the third column bet is 33.3%, assuming the ball lands in red, and ignoring both zeroes. However, by the same logic, if the ball lands in black, the probability of the third column bet winning is 4/18 = 2/9. A bet with a 2/9 chance of winning that pays 2 to 1 would have a house advantage of 33.33%. So, looking at it your way, the third column bet would have a player advantage of 33.33% half the time and a house edge of 33.33% the other half. Cancelling each other out, that bet has 0% house edge. Finally, mix in the two zeros and the overall player edge is (2/38)*-100% + (36/38)*0% = -5.26%.

Looking at it another way, the following table shows the number of ways all possible outcomes can occur, the probability, units won, and the contribution to the return. The lower right cell shows an an expected loss of -0.105263 units. Divided by the two units bet, that comes to a house edge of 5.26%.

 

Here is my solution (PDF).

This question is asked and discussed in my forum at Wizard of Vegas.

Yes, such a wheel is marketed by TCS John Huxley. I am not aware of any placements yet.

The formula for the house edge in standard roulette is z/(36+z), where z = number of zeroes. In this case, the house edge is 4/(36+4) = 4/40 = 1/10 = 10%.