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Las Vegas: How Physicists Used Science To Beat The Odds At Roulette Post written by Paul Halpern Physicist Paul Halpern is author of : How Richard Feynman and John Wheeler Revolutionized Time and Reality.
But not every Roulette wheel was created equally.
Indeed, scientists sometimes fall prey to the jangling siren song of clinking cash that lures so many people into casinos.
Here, thousands of simulations come together to calculate pi.
On some level, he surmised, knowing the initial state of a quantum system with absolute precision, as well as all of its relevant forces, would enable calculation of all of its future states.
On the contrary, untold experiments, drawing on the famous theorem of John Bell, have proven that quantum measurements possess a fundamental, built-in randomness that cannot be avoided.
Surely, the moment that a roulette wheel is spun and a ball is set into motion along a track in the opposite direction, the fate of which slot the ball will land in is sealed.
Moreover, the very fact that a tiny variation best casino in miami beach the initial conditions, such as the speed or trajectory of the ball, can dramatically alter the outcome, making predicting roulette outcomes a daunting task, in a practical sense.
Some casinos forbid bets after the wheel starts spinning.
Others�perhaps to build excitement�permit wagers for just a few rotations before the dealer calls no more bets.
Tantalizingly, but not realistically, one might hope that a quick glance at the spinning rotor might spur a solid guess and an auspicious placement of chips.
Hopping on their motorcycles, they scooted down to Reno, where they carefully studied the properties of roulette wheels to look for weaknesses to exploit.
Later, they also frequented casinos in Las Vegas to do the same.
Such flaws, the students realized, offered the key to successful prediction.
By studying the mechanical idiosyncrasies of various machines, they developed predictive models, carefully placed bets, and manage to win thousands of dollars.
They used much of their earnings to buy a boat and sail around the world.
Once his gambling ventures were finished, Hibbs, who had received his undergraduate degree from Caltech, returned there as a graduate student in theoretical physics, where he completed his PhD work under Richard Feynman.
He collaborated with Feynman on a co-authored textbook.
Hibbs and Feynman would remain close friends for life.
Evolving Lens Booksellers Hibbs and Walford were very open about their casino-beating methods.
In December 1949, they were featured prominently in a Life Magazine story.
Soon, and perhaps as a result, casinos began upgrading their equipment.
Biased wheels were replaced with ones that ran far more smoothly.
NBC-TV, 1959 Both Hibbs and Walford went on to illustrious careers.
Notably, he played a major role in the launch of Explorer 1, the first American satellite to orbit the earth, on January 31, 1958, six weeks after Sputnik was launched by the Soviets.
He also trained to be an astronaut in the Apollo mission, but was never selected to fly before the program was cancelled in the early 1970s.
NASA Walford, as a physician, became known for his advocacy of a severely calorie-restricted diet as the key to longevity.
Given that roulette wheels no longer had discernible defects, he realized he would need to develop a new strategy.
The key, he decided, would be a small computer, worn by https://clearadultskin.com/best/best-florida-casinos.html observing how the wheel was spun and the ball was launched, that was fast and powerful enough to calculate their trajectories and make a prediction.
Pretending just to be a casual onlooker, that observer would transmit the forecast by radio to a second participant charged with placing bets.
Toe switches would activate the computer once the wheel and ball were set into motion, collecting timing data for both.
Once the computer calculated the most likely best bet playing roulette, it would transmit that value as musical tones to a tiny speaker lodged in an earpiece.
The wires were camouflaged as much as possible.
For several years, on various occasions, Best bet playing roulette and Shannon, along with their wives, tried out their methods in Las Vegas.
Therefore, they had to mix their optimal bets, made after the wheel had revolved a few times and could be tracked, with random ones, best bet playing roulette before the spinning best odds in russian roulette table />That mixture made it hard for them to profit.
Finally, in 1966, Thorp decided to let the cat out of the bag and publish their methods.
By then, Thorp had became a professor of mathematics at U.
Irvine, a position he held for many years.
He continued to write books and articles about the mathematics and science of gambling.
Led by astrophysics major J.
Thomas Bass, another group member, would document their antics in a popular book called.
To eliminate the need for extra wiring, the computer would send its prediction as a direct signal to part of the foot, a bit like the vibrating mechanisms of modern phones.
They managed to use their system discretely best bet playing roulette casinos, estimating a 44% profit on each dollar bet.
All earnings were pooled by the group.
Photo of The Eudaemonic Pie display at the Heinz Nixdorf Museum.
By the mid-1980s, casinos upgraded their equipment once more and prohibited the use of computers.
Although, in theory, it would now be harder for students today to gain an advantage through their knowledge of mathematics and physics, the so-called MIT Blackjack Team implemented a system to great success in the 1990s, as detailed in the book.
From Hibbs and Walford, to Thorp and Shannon, and finally to Farmer and Packard, we see how an interest in beating the odds would lead to successful scientific careers.
On the click wheel of life, their gambles paid off splendidly.
Starts With A Bang is dedicated to exploring the story of what we know about the Universe as well as how we know it, with a focus on physics, astronomy, and the� Read More Starts With A Bang is dedicated to exploring the story of what we know about the Universe as well as how we know it, with a focus on physics, astronomy, and the scientific story that the Universe tells us about itself.

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Wouldn't you have like a 70% of winning?
The same house edge as on every bet in double-zero roulette except the 0-00-1-2-3 combination, which is 7.
I'm not talking about the long run; just a one-time bet.
If you want to increase your chances of winning even more then bet on any 35 numbers.
The probability of winning will be 92%.
Is there a way of combining bets in roulette to maximize one's odds?
For example, a dozen bet pays out 2 to 1.
If I place two dozen bets, say the first and second set of 12, I have a 63.
Although I really only gain 1 to 1 rather than 2 to 1 if I win, since part of my bet has to lose since the winning number cannot be in both the first and second set of twelvesthe odds have been slightly shifted in my favor by combining two bets.
Have the odds on these sorts of combinations been determined?
If they have been, where might I be able to find them?
There are ways to increase your probability of winning, but at the cost of winning less relative to your total wager.
I was playing roulette last nigh using the "Martingale" method of doubling down twice after the 1st loss.
Dumb, I know, but I usually don't lose much and I gamble a long time.
Anyway, what ended the game for me was I was betting even, and in four rolls the number 9 came up in three of the spins.
What are the odds of that?
Does that sound suspicious?
For that matter, have casinos ever been caught cheating?
However, if you play long enough you almost can't help but notice unusual events like this.
This does not nearly rise to the level of being suspicious.
Cheating does occur in real casinos.
It is usually a rogue dealer who is caught by casino security.
There have been some strong cases of cheating made against online casinos but no governmental authority has ever convicted anyone to the best of my knowledge.
Isn't it an even worse roulette betting strategy to bet multiple numbers on the inside during one bet as most players do vs.
It seems to me that "hedging" just guarantees that certain in the above case 9 bets will ALWAYS lose?
You don't address "hedging" on your page?
The sixth commandment is "never hedge thy bets.
Both these methods have the same total expected return of 94.
I read your page on systems and I have been telling people this for years!
I deal roulette in a casino and I have seen all of the systems at one time or another.
That means that I have seen it win more than lose.
This covers all but six numbers 19 through 22 and will yield a 15 dollar payout every time the ball misses those 6 numbers EXCEPT when 0 or 00 hits in which case it's 40 dollars.
I know it sounds nuts!!!
But trust me, I'm here to tell you I have seen it win more than lose.
It also works in reverse duh.
In your observations you have likely seen fewer than expected 19-24 occurrences, which accounts for the illusion that this method is winning.
Say, Wizard I was wondering about roulette.
With the boss media software, you can spin the wheel without actually placing a bet.
Isn't this in the player's favor?
With some of the roulette betting systems I have seen, like the Martingale you double up when you lose and such.
Can't you just watch the wheel without placing a bet and based on the previous spins place your bet.
An example would be to spin the wheel 5 times before placing a bet.
Suppose that all the numbers were odd.
Wouldn't it make sense to begin playing the wheel by betting on even?
I know this kinda buys in to the gambler's fallacy and the wheel doesn't care what the last spin was, but also the probability diminishes with each spin that an odd number will continue to come up.
Am I on to something or just going over an old theory?
You're just rehashing the gambler's fallacy.
If the https://clearadultskin.com/best/texas-holdem-raise-after-all-in.html landed in odd 100 times in a row on a fair wheel the odds that the next spin would be even are still the same as every spin, 47.
So it does not help that you can spin without betting.
The ball does not have a memory.
While Roulette clearly cannot be beaten by chance, I have heard that it can be beaten by physics two ways in theory.
Way one: a high tech device, which measures the velocity of the ball against the velocity of the wheel and predicts the outcome sector of the wheel with like a 40% accuracy.
Way two: Wheel bias.
Obviously a wheel would have to have a bias of at least 5.
The question is, how many spins would you say, Wizard, does it take to determine wheel bias, if there is any?
I've heard of both these techniques being used.
I don't know much about devices to clock wheels, except they are known to exist and be used from time to time.
Here in Nevada such a device would be highly illegal.
Taking advantage of biased wheels I hear a lot more about.
It has been done lots of times.
I think casinos with old wheels are the most vulnerable target.
I've been saying for years that I think Argentina is a ripe target for that.
When playing the Martingale double-up system against the single-zero roulette wheel on any one of the even chances.
I figured that you will lose one time in every 248 sessions.
Meaning a session that runs to completion with either a win of one unit or a loss of 255 units.
Am I figuring correctly, if not could you please give the correct odds?
If the maximum loss is 255 units then you can bet up to 8 times.
So, you have 99.
On average, in single-zero roulette, how often will a number repeat ex.
You can expect a repeat once every 37 pairs of numbers.
So, with 36 numbers we have 35 pairs of numbers.
Everybody says that roulette cannot be beaten in the long run with mathematical systems.
But, how can you explain the fact that there are professional gamblers who make their living at roulette?
I don't think it's just boasting.
They actually win more than they lose with everyday playing.
Show me someone who is winning at a fair game of roulette and I'll show you somebody who is just lucky, and will likely lose it all back.
You can only best bet playing roulette beat roulette by advantage play, like exploiting a biased wheel, or clocking the wheel.
I have read everything you have to say about roulette, but see nothing about roulette spinners.
Under the watchful eye of the pit boss the roulette spinner throws "sections" to improve the odds of the house.
The spinner that can throw greens, not every time but with good percentages of time, maybe one out of 7 or 8 when he wants to wipe out a large progressive better.
Furthermore casinos have nothing to fear from progressive bettors.
Most of the time progressive bettors win, but the few that hit their bankroll limits pay for all the winners and then some more for the casino.
Furthermore it would take a great deal of skill to deliberately spin a ball into a specific section.
Thus, the expected probability of coming out ahead over 1, 100, visit web page 10000 trials, respectively is 0.
Is my analysis correct?
I assumed normality Thanks!
Thanks for the compliment.
First of all the standard deviation on any even money bet is 0.
The probability of coming out ahead by flat betting even money bets over 1, 100, and 10000 spins is 0.
The probability of coming out ahead by flat betting single number bets over 1, 100, and 10000 spins is 0.
It seems you are trying to argue that single number bets are better because of the higher probability of finishing ahead over click the following article bets.
This is true, however the probability of a substantial loss is also much greater.
Over a session the expected results always fall somewhere on a bell curve.
With high volatility bets like single numbers the bell curve is wide, allowing for a much wider range of net results, both good and bad.
Is he just lucky or is there any system that works?
I read your topic in Roulette on the Martingale method.
Because black came up 8 times in a row.
Do you think this method would work in a casino?
Like I said black came up 8 times in a row.
But do you think that the player hand would win best betting odds for blackjack times in a row?
Plus this game is good because a tie is a push, where in roulette 0, or 00, is a loss.
The Martingale is dangerous on every game and in the long run will never win.
However it is better to use in baccarat than roulette, just because of the lower house edge.
The probability of the player winning 8 times in a row is 0.
Also keep in mind you could win a hand late in the series and still come out behind because of the commission.
You say all betting systems will fail.
If you only play for one spin and want to maximize your probability of winning then bet equally on 35 of the numbers.
The system was tested and developed in conjunction with" Spin roultte Gold" by Frank Scoblete and "Roulette system tester" by Eric St.
I mean, beating these testbooks have to mean something.
And do you think I should try them out??
Best wishes 7500 spins?
Anyone can show a profit of 7.
Same is true about 15000 spins.
Most systems are designed to have a lot of small wins and small number of large losses.
A system requiring a huge bankroll can easily go 15000 spins and show a profit.
Eventually the losses will come in and it won't pass the test of time.
The big losses might also come at the beginning.
The true way to put a system to the test is to play it over billions of trials.
My opinion about these systems is the same as all systems, they best bet playing roulette worthless.
I have no problem with you trying them out but I do have a problem with anyone putting one dime in the pockets of those selling them.
Note: See the follow up to this question in the.
However, should you lose you can now bet187.
Now should you lose on both spins you still have 187.
Hence, for the SAME capital and for the SAME payoff you are able to increase your PROBABILITY of success as in option B if you play fewer numbers with less money but for MAYBE more spins.
Assuring you of my highest regards and awaiting the favor of your reply I remain.
You are correct that option B has the greater probability of success, although the goal and the capital are the same.
The reason is the average amount bet in option B is less, thus your money is exposed to the house edge less, thus the probability of winning increases.
This way I was not necessarily exposing my entire stake to the house edge, which increased my probability of winning.
I measure the value of a bet to be the expected return, not the probability of winning.
Although betting all 38 numbers has a 0% chance of showing a net win, the down side is losing only 5.
If forced to bet and you want to minimize variance then you should bet all 38 numbers.
So to answer your question there is no optimal range of numbers.
All ranges are equal in expected value.
There is today about a British man who will bet his life savings on one roulette roll.
My friend and I have been debating about what the best casino bet is for this type of wager.
If you can only place one bet, and you wish to maximize your odds, what is the best game to play and what is the best bet?
First, let me say this guy was a fool.
However had he taken a 10 minute ride to the Bellagio, Mirage, or Aladdin he could have made the bet on a single zero wheel which follows the European rule of giving half an even money bet back if the ball lands in zero.
He planned to make an even money bet anyway.
So, at these wheels with full European rules his house edge would have been only 1.
To answer your question, if forced to make just one even money type bet I would have chosen the banker bet in baccarat with a house edge of 1.
Risk of ruin questions are mathematically complicated.
I live in NJ about two hours north of Atlantic City.
Do you have an idea as to where the closest European Roulette Wheel to someone in my part of the country is?
There are lots of single zero wheels in Atlantic City.
I was at Casino On Net.
I was playing Roulette.
I spun the wheel 5 times without betting, waiting for a pattern of one of the sets to not come up so I could bet on it, hoping this would shift probability of it landing in my favor.
I kept betting in L, I figured the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 OR 12 would land within 12 Spins so that I could at least recoup my money.
Congratulations on a great site.
I totally understand your anger over the spread of 6 to 5 Blackjack payouts but am very curious as to why Americans seem best bet playing roulette accept 00 Roulette without any argument.
You make a valid point.
The house edge in 6 to 5 blackjack is 1.
However I have learned through the years that it is almost hopeless getting players to leave a game they like, regardless of how bad the house edge is.
So the best I can do is advise them how to play their game of choice.
For blackjack players there is still no shortage of 3 to 2 games out there.
Playing 6 to 5 is giving the casino an extra 0.
I also stress the importance of looking for single-zero roulette if you are a roulette player.
So I see no inconsistency.
Can you please explain to me how the table limits for roulette works and what is the difference between minimum limit for individual number and table.
If possible, please give examples.
There are usually two minimums in roulette.
Outside bets are all even money bets, column bets, and dozen bets.
Inside bets are those on the numbers, including groups of 2, 3, 4, 5, and 6.
I work in a casino and have a bet that says a roulette dealer cannot influence the outcome of a roll.
There are definitely those who think it can be done.
Not to a number of course, but perhaps a section of the wheel.
What would you consider a good test to reasonably determine whether a dealer has influenced the results?
Assuming the number of trials is reasonable for us to attempt I will gladly share the results.
If this could be done then dealers could easily conspire with players and share in the profits.
Yet I never hear of this happening.
A good test would be to get somebody who claims to be able to influence the roll and have him attempt to land it in a particular half of the wheel as many times as possible over 100 spins.
The more times he makes it the greater weight his claim will have.
The table below shows the probability of 50 to 70 successful spins.
For example, the probability of 60 or more successful spins is 2.
Common confidence thresholds in statistics are the 90%, 95%, and 99% levels.
To beat a 90% confidence test, in which the probability of failing given random spins is 90%, the number of successful spins would need to be 57 or more.
To beat a 95% test the number would need to be 59 or more, and at 99% the number would need to be 63 or more.
Wins Probability 70 0.
Are there any systems that would be effective since there is no zero?
Without the zero could one effectively play both black and red at the same time since there is no fear of the zero?
I tried to find that game but the site was down when I checked.
However, assuming such a game did exist, the answer is no.
No system could be expected to beat it, nor lose to it, over the long-run.
The expected value of every system would be exactly zero.
What is the probability of having any one number 0-36 come up more than once in three consecutive spins?
Thanks for you time!
I think you may be interested in this.
I use Betfair here in the UK.
I am sure you nice people in America are not allowed to use it for some reason, anyway if you are not familiar with it go to betfair.
It is a betting exchange, not a bookmakers.
Anyway, my question is this.
They are now offering roulette with NO ZERO, yes no zero.
It is genuine really.
Can you think of a good strategy one could use, and if so would you keep it to yourself?
They offer other casino games with no house edge too.
I tried to sign up for an account there to check this out but they block U.
Even in a zero house edge game like no-zero roulette there is still no betting system that will get above, or below, that 0% figure.
No matter what you do the more you do of it the closer the actual house win will get to 0%.
Gonzalo Garcia-Pelayo and his family won a lot of money in many casinos all over the world.
They even published a book, and describe how they did it.
I saw a television show about him once, and I applaud what he did.
What successfully did was survey how often the ball landed in each number, in an effort to find, and then exploit, biased roulette wheels.
This I would call a strategy, as opposed to a system.
There are lots of profitable strategies for beating the casinos, but zero profitable betting systems.
If I show you a Roulette game with no zeros and all usual roulette rules apply, is it possible to win 100% of the time?
It just has number 1-36, and all the standard roulette rules apply.
Do you see any way to take advantage of this?
There must be a money management system that could work profitably with these table limits.
Any advice is appreciated.
Thanks for the kind words.
Hello, for obvious reasons I would appreciate if you not share my name with anyone.
At the casino where I work, there is virtual unanimity amongst the roulette dealers that they can "control the spin" and hit sections of the wheel with ease or miss other sections on purpose.
Given all the factors in the spin of a ball in roulette, including the canoes bumps on the side, direction of ball and speed of wheel, etc.
Would it be enough that a crooked dealer could help a player overcome the huge house advantage?
This is far from the first time I have heard this claim, go here I am very skeptical of it.
Most dealers also believe myths like a bad third-baseman will cause the other players to lose in best casino in milano, so as a group they are not the most skeptical bunch.
What I think is happening is they remember the times they were successful at attempting to control the spin, and conveniently forget the times they were not.
If dealers really could do this, it would be easy to have a confederate play, causing him to win, and causing other players to lose, to make up for it.
Yet, you never hear about this happening.
I suppose the believers could say that those doing it are just keeping a low profile, but that is what believers in worthless betting systems say too.
If this were as easy as the roulette dealers where you work claim, the cheating problem as a result would be rampant.
In double-zero roulette, what is the probability that any number will not have hit by the 200th spin?
The reason this is incorrect is it double counts two numbers not being hit.
So we need to subtract those probabilities out.
We need to subtract the probability of avoiding both numbers.
However, now we have canceled out the probability of three numbers not hitting.
For any given group of three numbers, we triple counted the probability of any single pokerblog not being hit.
We then triple subtracted for each way to choose two numbers out of the three, leaving with zero for the probability that all three numbers were not hit.
Yet, now we have over-counted the probability of four numbers not hitting.
Then we added back in the 4 groups of 3 out of the 4.
To adjust for the double counting we must subtract for each group.
Continuing in the process we would keep alternating adding and subtracting, all the way until missing 37 numbers.
Numbers Hit Observations Ratio 31 or Less 0 0 32 1 0.
I have a friend who was part of a casino staff who watched over roulette tables, and he told me that when people start to win the casino changes the croupier.
I have also seen a member of the staff ask a croupier to spin the roulette wheel at a different speed.
Sadly, ignorance can go pretty high up the ladder.
However, that issue aside, changing dealers does not change the odds.
There is no such thing as a lucky or an unlucky dealer.
Superstition is a difficult thing to let go of.
As I have said many times, the more ridiculous a belief is, the more tenaciously article source tends to be held.
Dear sir, I "clocked" an automated single-zero roulette game for 8672 games.
My predetermined number came up an amazing 278 times.
I chose the number because of the wear and tear of the pocket.
If my terminology is correct, "clocking a wheel" means to predict where the ball will land judging by the ball speed, ball location, and wheel speed.
It sounds like what you are doing is exploiting a biased wheel, which is a different advantage play.
This allows the player to predict where the ball will land based on ball location and past results.
The standard deviation is the square root of the variance, or 15.
The reason for subtracting 0.
Suffice it to say it is an adjustment factor for using a continuous function to estimate a discrete function.
Doing a Gaussian approximation, the probability of hitting your number that many times, or more, is 0.
So, there is a good chance you found a biased wheel.
However, there is still a 1 in 466 chance it was just good beste strategie roulette />There is a well-known story about a freeze-out competition between a Japanese high roller, Kashiwagi, and Donald Trump, that took place 20 years ago.
Assume that Kashiwagi always bet the maximum on Banker.
What is the probability that Kashiwagi will win?
The math works out more easily if he bet on the Player.
are best casino gambling stories simply work out a similar problem in roulette at my mathproblems.
It is much more complicated on the Banker bet, because of the 5% commission.
That would result in the distinct possibility of the player overshooting his goal.
This question was raised and discussed in the forum of my companion site.
Counting the first trial, I show the mean is 8.
So the probability of a repeat within 8 numbers is 100% - 45.
I suspect most people would estimate that that probability of a repeat within 8 numbers would be less than that.
So you would be betting on 8 or fewer, and your friend 9 or more.
Basically, whichever side covers the median of 8 is likely to win.
This question was raised and discussed in the forum of my companion site.
No, he is not correct.
In all bet calculations, round down.
In other words, always try to reach the goal, with just one bet, if you can, without exceeding the goal.
If there are multiple ways to accomplish this, then go with the one with the greatest probability of winning.
What about other games, you might ask?
According to the Discovery Channel voice-over guy, "Everyone agrees that roulette is the best get rich quick scheme in the casino.
Even limiting ourselves to common games and rules, I find craps to be better.
In particular, betting the don't pass and laying the odds.
This assumes the player may lay 6x odds, regardless of the point which is the case when 3x-4x-5x odds are allowed taking the odds.
This probability of success is 0.
In that case, the probability of success using my Hail Mary strategy is 2.
In both cases, greater than the 2.
In all fairness, the Discovery Channel would have never put the insane rant above on the air and was surely looking for something simple that the masses would understand.
Andy was surely giving them something they wanted to hear.
The basic premise of his advice is that if you want to reach a certain goal, then click the following article hit-and-run strategy is much better than letting the house edge grind you down with multiple bets.
That is definitely true and something I've been preaching for 17 years.
Wizard's "Hail Mary" strategy for craps.
In calculating bets, never bet so much that you overshoot the goal.
Also, never make a bet amount that will cause you to get rounded down.
Otherwise, lay whatever you can.
So, I hope Andy and the Discovery Channel are happy.
I've spent days running simulations to prove them wrong.
This question was raised and discussed on my forum at.
What is the probability of the ball landing in 1, 2, and 3 within 4, 5, 6, 7, 8, or 9 spins in roulette?
The general formula is: Pr Ball lands in 1 + Pr Ball lands in 2 + Pr Ball lands in 3 - Pr Ball lands in 1 and 2 - Pr Ball lands in 1 and 3 - Pr Ball lands in 2 and 3 + Pr Ball lands in 1, 2, and 3.
The following table shows the probability of rolling all three numbers for various number of spins from 3 to 100 for click the following article and double-zero roulette.
Spins Single Zero Double Zero 3 0.
If bets on these outcomes paid the usual 35 to 1, then the house edge would be exactly 0%.
The house edge on any bet on the numbers 1 to 36 would be 5.
Compare that to the 5.
My advice in this game would be to bet the zero and double-zero only.
If anyone can confirm or deny https://clearadultskin.com/best/10-best-starting-poker-hands.html rules and pays, please let me know.
I recorded 7,456 spins in roulette.
The results are as follows.
I suspect the wheel is biased but am not sure if the data is conclusive enough to play it.
Winning Number Occurences 0 204 28 214 9 175 26 177 30 203 11 181 7 223 20 205 32 184 17 222 5 224 22 241 34 194 15 210 3 209 24 176 36 203 13 217 1 217 00 197 27 173 10 195 25 198 29 217 12 197 8 207 19 163 31 180 18 201 6 186 21 203 33 171 16 164 4 200 23 191 35 163 14 177 2 194 Total 7456 The following graph shows your results in sequential order on the wheel.
The blue line shows your results.
The red line is the number you need, 207.
A chi-squared test on this distribution comes back with a statistic of 68.
The probability of a result this skewed or more is 1 in 725.
I don't think the chi-squared is the perfect test for this situation because it doesn't consider the ordering of the outcomes, but don't know of a better test.
Some have suggested thebut I don't think that is appropriate.
If there are any other appropriate tests, I'm all ears.
I can say if you had bet the 3-number arc around the number 5, you would have best bet playing roulette a 10.
However, if you increased that to a 7-number arc, the advantage drops to 2.
If forced to an answer in plain simple English, I would say the wheel shows evidence, but not proof beyond a reasonable click here, that the wheel is biased.
However, that bias is probably not enough to significantly and confidently overcome the house edge.
Assuming the casino doesn't switch around the wheels among the tables, I would say that more data should be collected before betting large amounts of money.
I'm sorry this answer is so noncommittal.
This question is raised and discussed in my forum at.
In single-zero roulette, what is the mean and median number of spins required for every number to appear at least once?
Answering the mean is much easier, so we'll start with that.
The median is much more complicated.
To find the exact answer, as opposed to using a random simulation, one needs to use a lot of matrix Algebra.
I've discussed how to solve similar problems in other Ask the Wizard questions, so I won't go through the details again.
One example of a this web page question is the one on getting a 6-6 pair in the hole three times in a row, as discussed in.
Suffice it to say that the probability of seeing every number in 145 spins is 0.
Thus, the median is 146.
This question is asked and discussed in my forum at.
If a ball landed in red the last 20 spins in roulette, what is the probability it will land in black the next spin?
I think you're wrong about the previous question.
The odds must overwhelmingly favor black.
That's true, but it doesn't matter.
The fact is the past doesn't matter in games of independent trials like roulette.
I've thought of a way to beat the casinos in roulette!
Start with a small wager on any even-money bet, like red or black.
If it loses, then double the bet on the same thing.
Then keeping doubling until it wins.
The winning outcome has to happen eventually and when it does I'll profit my original wager.
What is your opinion?
Also, please don't tell anybody.
This is probably the most popular of all betting systems, known as the Martingale.
Gamblers have been conceiving of it and using it since time immemorial.
Like all betting systems, not only doesn't it beat the house advantage, it doesn't even dent it.
The reason is the gambler will eventually have a bad losing streak where his bankroll isn't enough to make another double.
In your previous answer, you explained why the Martingale doesn't work.
Then how about the opposite, doubling your bet after each win until a desired target is hit?
This is known as the anti-Martingale and is equally worthless.
The times your bankroll gets grinded down to nothing will outweigh the winnings when you hit your target.
Regardless of what betting system you use, or none at all, the more you play, the more your ratio of more info lost to money bet will approach 5.
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Do You Know the Best and Worst Roulette Bets?
Do You Know the Best and Best bet playing roulette Roulette Bets?
The outcome of every spin of the roulette wheel is independent of any other and is not influenced by prior spins.
Even if you best bet playing roulette ten best bet playing roulette numbers after ten spins, the chance of getting red in the next spin is the same as to get black.
The wheel is divided into either 37 or 38 slots, depending on whether you are playing the European version single zeroor the American version which has an extra space in the double zero slot.
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Bets to Place If you plan to play Roulette, stick with the bets whose odds almost mirror their payouts.
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Wouldn't you have like a 70% of winning?
The same house edge as on every bet in double-zero roulette except the 0-00-1-2-3 combination, which is 7.
I'm not talking about the long run; just a one-time bet.
If you want to increase your chances of winning even more then bet on any 35 numbers.
The probability of winning will be 92%.
Is there a way of combining bets in roulette to maximize one's odds?
For example, a dozen bet pays out 2 to 1.
If I place two dozen bets, say the first and second set https://clearadultskin.com/best/best-3d-poker-games-to-play.html 12, I have a 63.
Although I really only gain 1 to 1 rather than 2 to 1 if I win, since part of my bet has to lose since the winning number cannot be in both the first and second set of twelvesthe odds have been slightly shifted in my favor by combining two bets.
Have the odds on these sorts of combinations been determined?
If they have been, where might I be able to find them?
There are ways to increase your probability of best bet playing roulette, but at the cost of winning less relative to your total wager.
I was playing roulette last nigh using the "Martingale" method of doubling down twice after the 1st loss.
Dumb, I know, https://clearadultskin.com/best/best-casino-to-play-in-detroit.html I usually don't lose much and I gamble a long time.
Anyway, what ended the game for me was I was betting even, and in four rolls the number 9 came up in three of the spins.
What are the odds of that?
Does that sound suspicious?
For that matter, have casinos ever been caught cheating?
However, if you play long enough you almost can't help but notice unusual events like this.
This does not nearly rise to the level of being suspicious.
Cheating does occur in real casinos.
It is usually a rogue dealer who is caught by casino security.
There have been some strong cases of cheating made against online casinos but no governmental authority has ever convicted anyone to the best of my knowledge.
Isn't it an even worse roulette betting strategy to bet multiple in best tahoe casino on the inside during one bet as most players do vs.
It seems to me that "hedging" just guarantees that certain in the above case 9 bets will ALWAYS lose?
You don't address "hedging" on your page?
The sixth commandment is "never hedge thy bets.
Both these methods have the same total expected return of 94.
I read your page on systems and I have been telling people this for years!
I deal roulette in a casino and I have seen all of the systems at one time or another.
That means that I have seen it win more than lose.
This covers all but six numbers 19 through 22 and will yield a 15 dollar payout every time the ball misses those 6 numbers EXCEPT when 0 or 00 hits in which case it's 40 dollars.
I know it sounds nuts!!!
But trust me, I'm here to tell you I have seen it win more than lose.
It also works in reverse duh.
In your observations you have likely seen fewer than expected 19-24 occurrences, which accounts for the illusion that this method is winning.
Say, Wizard I was wondering about roulette.
With the boss media software, you can spin the wheel without actually placing a bet.
Isn't this in the player's favor?
With some of the roulette betting systems I have seen, like the Martingale you double up when you lose and such.
Can't you just watch the wheel without placing a bet and based on the previous spins place your bet.
An example would be to spin the wheel 5 times before placing a bet.
Suppose that all the numbers were odd.
Wouldn't it make sense to begin playing the wheel by betting on even?
I know this kinda buys in to the gambler's fallacy and the wheel doesn't care what the last spin was, but also the probability diminishes with each spin that an odd number will continue to come up.
Am I on to something or just going over an old best bet playing roulette />You're just rehashing the gambler's fallacy.
If the ball landed in odd 100 times in a row on a fair wheel the odds that the next spin would be even are still the same as every spin, 47.
So it does not help that you can spin without betting.
The ball does not have a memory.
While Roulette clearly cannot be beaten by chance, I have heard that it can be beaten by physics two ways in theory.
Way one: a high tech device, which measures the velocity of the ball against the velocity of the wheel and predicts the outcome sector of the wheel with like a 40% accuracy.
Way two: Wheel bias.
Obviously a wheel would have to have a bias of at least 5.
The question is, how many spins would you say, Wizard, does it take to determine wheel bias, if there is any?
I've heard of both these techniques being used.
I don't know much about devices to clock wheels, except they are known to exist and be used from time to time.
Here in Nevada such a device would be highly illegal.
Taking advantage of biased wheels I hear a lot more about.
It has been done lots of times.
I think casinos with old wheels are the most vulnerable target.
I've been saying for years that I think Argentina is a ripe target for that.
When playing the Martingale double-up system against the single-zero roulette wheel on any one of the even chances.
I figured that you will lose one time in every 248 sessions.
Meaning a session that runs to completion with either a win of one unit or a loss of 255 units.
Am I figuring correctly, if not could you please give the correct odds?
If the maximum loss is 255 units then you can bet up to 8 times.
So, you have 99.
On average, in single-zero roulette, how often will a number repeat ex.
You can expect a repeat once every 37 pairs of numbers.
So, with 36 numbers we have 35 pairs of numbers.
Everybody says that roulette cannot be beaten in the long run with mathematical systems.
But, how can you explain the fact that there are professional gamblers who make their living at roulette?
I don't think it's just boasting.
They actually win more than they lose with everyday playing.
Show me someone who is winning at a fair game of roulette and I'll show you somebody who is just lucky, and will likely lose it all back.
You can only skillfully beat roulette by advantage play, like exploiting a biased wheel, or clocking the wheel.
I have read everything you have to say about roulette, but see nothing about roulette spinners.
Under the watchful eye of the pit boss the roulette spinner throws "sections" to improve the odds of the house.
The spinner that can throw greens, not every time but with good percentages of time, maybe one out of 7 or 8 when he wants to wipe out a large progressive better.
Furthermore casinos have nothing to fear from progressive bettors.
Most of the time progressive bettors win, but the few that hit their bankroll limits pay for all the winners and then some more for the casino.
Furthermore it would take a great deal of skill to deliberately spin a ball into a specific section.
Thus, the expected probability of coming out ahead over 1, 100, and 10000 trials, respectively is 0.
Is my analysis correct?
I assumed normality Thanks!
Thanks for the compliment.
First of all the standard deviation on any even money bet is 0.
The probability of coming out ahead by flat betting even money bets over 1, 100, and 10000 spins is 0.
The probability of coming out ahead by flat betting single number bets over 1, 100, and 10000 spins is 0.
It seems you are trying to argue that single number bets are better because of the higher probability of finishing ahead over multiple bets.
This is true, however the probability of a substantial loss is also much greater.
Over a session the expected results always fall somewhere on a bell curve.
With high volatility bets like single numbers the bell curve is wide, allowing for a much wider range of net results, both good and bad.
Is he just lucky or is there any system that works?
I read your topic in Roulette on the Martingale method.
Because black came up 8 times in a row.
Do you think this method would work in a casino?
Like I said black came up 8 times in a row.
But do you think that the player hand would win 8 times in a row?
Plus this game is good because a tie is a push, where in roulette 0, or 00, is a loss.
The Martingale is dangerous on every game and in the long run will never win.
However it is better to use in baccarat than roulette, just because of the lower house edge.
The probability of the player winning 8 times in a row is 0.
Also keep in mind you could win a hand late in the series and still come click behind because of the commission.
You say all https://clearadultskin.com/best/best-home-poker-games.html systems will fail.
If you only play for one spin and want to maximize your probability of winning then bet equally on 35 of the numbers.
The system was tested and developed in conjunction with" Spin roultte Gold" by Frank Scoblete and "Roulette system tester" by Eric St.
I mean, beating these testbooks have to mean something.
And do you think I should try them out??
Best wishes 7500 spins?
Anyone can show a profit of 7.
Same is true about 15000 spins.
Most systems are designed to have a lot of small wins and small number of large losses.
A system requiring a huge bankroll can easily go 15000 spins and show a profit.
Eventually the losses will come in and it won't pass the test of time.
The big losses might also come at the beginning.
The true way to put a system to the test is to play it over billions of trials.
My opinion about these systems is the same as all systems, they are worthless.
I have no problem with you trying them out but I do have a problem with anyone putting think, best online casino for craps apologise dime in the pockets of those selling them.
Note: See the follow up to this question in the.
However, should you lose you can now bet187.
Now should you lose on both spins you still have 187.
Hence, for the SAME capital and for the SAME payoff you are able to increase your PROBABILITY of success as in option B if you play fewer numbers with less money but for MAYBE more spins.
Assuring you of my highest regards and awaiting the favor of your reply I remain.
You are correct that option B has the greater probability of success, although the goal and the capital are the same.
The reason is the average amount bet in option B is less, thus your money is exposed to the house edge less, thus the probability of winning increases.
This way I was not necessarily exposing my entire stake to the house edge, which increased my probability of winning.
I measure the value of a bet to be the expected return, not the probability of winning.
Although betting all 38 numbers has a 0% chance of showing a net win, the down side is losing only 5.
If forced to bet and you want to minimize variance article source you should bet all 38 numbers.
So to answer your question there is no optimal range of numbers.
All ranges are equal in expected value.
There is today about a British man who will bet his life savings on one roulette roll.
My friend and I have been debating about what the best casino bet is for this type of wager.
If you can only place one bet, and you wish to maximize your odds, what is the best game to play and what is the best bet?
First, let me say this guy was a fool.
However had he taken a 10 minute ride to the Bellagio, Mirage, or Aladdin he could have made the bet on a single zero wheel which follows the European rule of giving half an even money bet back if the ball lands in zero.
He planned to make an even money bet anyway.
So, at these wheels with full European rules his house edge would have been only 1.
To answer your question, if forced to make just one even money type bet I would have chosen the banker bet in baccarat with a house edge of 1.
Risk of ruin questions are mathematically complicated.
I live in NJ about two hours north of Atlantic City.
Do you have an idea as to where the closest European Roulette Wheel to someone in my part of the country is?
There are lots of single zero wheels in Atlantic City.
I was at Casino On Net.
I was playing Roulette.
I spun the wheel 5 times without betting, waiting for a pattern of one of the sets to not come up so I could bet on it, hoping this would shift probability of it landing in my favor.
I kept betting in L, I figured the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 OR 12 would land within 12 Spins so that I could at least recoup my money.
Congratulations on a great site.
I totally understand your anger over the spread of 6 to 5 Blackjack payouts but am very curious as to why Americans seem to accept 00 Roulette without any argument.
You make a valid point.
The house edge in 6 to 5 blackjack is 1.
However I have learned through the years that it is almost hopeless getting players to leave a game they like, regardless of how bad the house edge is.
So the best I can do is advise them how to play their game of choice.
For blackjack players there is still no shortage of 3 to 2 games out there.
Playing 6 to 5 is giving the casino an extra 0.
I also stress the importance of looking for single-zero roulette if you are a roulette player.
So I see no inconsistency.
Can you please explain to me how the table limits for roulette works and what is the difference between minimum limit for individual number and table.
If possible, please give examples.
There are usually two minimums in roulette.
Outside bets are all even money bets, column bets, and dozen bets.
Inside bets are those on the numbers, including groups of 2, 3, 4, 5, and 6.
I click in a casino and have a bet that says a roulette dealer cannot influence the outcome of a roll.
There are definitely those who think it can be done.
Not to a number of course, but perhaps a section of the wheel.
What would you consider a good test to reasonably determine whether a dealer has influenced the results?
Assuming the number of trials is reasonable for us to attempt I will gladly share the results.
If this could be done then dealers could easily conspire with players and share in the profits.
Yet I never hear of this happening.
A good test would be to get somebody who claims to be able to influence the roll and have him attempt to land it in a particular half of the wheel as many times as possible over 100 spins.
The more times he makes it the greater weight his claim will have.
The table below shows the probability of 50 to 70 successful spins.
For example, the probability of 60 or more successful spins is 2.
Common confidence thresholds in statistics are the 90%, 95%, and 99% levels.
To beat a 90% confidence test, in which the probability of failing given random spins is 90%, the number of successful spins would need to be 57 or more.
To beat a 95% test the number would need to be 59 or more, and at 99% the number would need to be 63 or more.
Wins Probability 70 0.
Are there any systems that would be effective since there is no zero?
Without the zero could one effectively play both black and red at the same time since there is no fear of the zero?
I tried to find that game but the site was down when I checked.
However, assuming such a game did exist, the answer is no.
No system could be expected to beat it, nor lose to it, over the long-run.
The expected value of every system would be exactly zero.
What is the probability of having any one number 0-36 come up more than once in three consecutive spins?
Thanks for you time!
I think you may be interested in this.
I use Betfair here in the UK.
I am sure you nice people in America are not allowed to use it for some reason, anyway if you are not familiar with it go to betfair.
It is a betting exchange, not a bookmakers.
Anyway, my question is this.
They are now offering roulette with NO ZERO, yes no zero.
It is genuine really.
Can you think of a good strategy one could use, and if so would you keep it to yourself?
They offer other casino games with no house edge too.
I tried to sign up for an account there to check this out but they block U.
Even in a zero house edge game like no-zero roulette there is still no betting system that will get above, or below, that 0% figure.
No matter what you do the more you do of it the closer the actual house win will get to 0%.
Gonzalo Garcia-Pelayo and his family won a lot of money in many casinos all over the world.
They even published a book, and describe how they did it.
I saw a television show about him once, and I applaud what he did.
What successfully did was survey how often the ball landed in each number, in an effort to find, and then exploit, biased roulette wheels.
This I would call a strategy, as opposed to a system.
There are lots of profitable strategies for beating the casinos, but zero profitable betting systems.
If I show you a Roulette game with no zeros and all usual roulette rules apply, is it possible to win 100% of the time?
It just has number 1-36, and all the standard roulette rules apply.
Do you see any way to take advantage of this?
There must be a money management system that could work profitably best bet playing roulette these table limits.
Any advice is appreciated.
Thanks for the kind words.
Hello, for obvious reasons I would appreciate if you not share my name with anyone.
At the casino where I work, there is virtual unanimity amongst the roulette dealers that they can "control the spin" and hit sections of the wheel with ease or miss other sections on purpose.
Given all the factors in the spin of a ball in roulette, including the canoes bumps on the side, direction of ball and speed of wheel, etc.
Would it be enough in craps way roll a crooked dealer could help a player overcome the huge house advantage?
This is far from the first time I have heard this claim, and I am very skeptical of it.
Most dealers also believe myths like a bad third-baseman will cause the other players to lose in blackjack, so as a group they are not the most skeptical bunch.
What I think is happening is they remember the times they were successful at attempting to control the spin, and conveniently forget the times they were not.
If dealers really could do this, it would be easy to have a confederate play, causing him to win, and causing other players to lose, to make up for it.
Yet, you never hear about this happening.
I suppose the believers could say that those doing it are just keeping a low profile, but that is what believers in worthless betting systems say too.
If this were as easy as the roulette dealers where you work claim, the cheating problem as a result would be rampant.
In double-zero roulette, what is the probability that any number will not have hit by the 200th spin?
The reason this is incorrect is it double counts two numbers not being hit.
So we need to subtract those probabilities out.
We need to subtract the probability of avoiding both numbers.
However, now we have canceled out the probability of three numbers not hitting.
For any given group of three numbers, we triple counted the probability of any single number not being hit.
We then triple subtracted for each way to choose two numbers out of the three, leaving with zero for the probability that all three numbers were not hit.
Yet, now we have over-counted the probability of four numbers not hitting.
Then we added back in the 4 groups of 3 out of the 4.
To adjust for the double counting we must subtract for each group.
Continuing in the process we would keep alternating adding and subtracting, all the way until missing 37 numbers.
Numbers Hit Observations Ratio 31 or Less 0 0 32 1 0.
I have a friend who was part of a casino staff who watched over roulette tables, and he told me that when people start to win the casino changes the croupier.
I have also seen a member of the staff ask a croupier to spin the roulette wheel at a different speed.
Sadly, ignorance can go pretty high up the ladder.
However, that issue aside, changing dealers does not change the odds.
There is no such thing as a lucky or an unlucky dealer.
Superstition is a difficult thing to let go of.
As I have said many times, the more ridiculous a belief is, the more tenaciously it tends to be held.
Dear sir, I "clocked" an automated single-zero roulette game for 8672 games.
My predetermined number came up an amazing 278 times.
I chose the number because of the wear and tear of the pocket.
If my terminology is correct, "clocking a wheel" means to predict where the ball will land judging by the ball speed, ball location, and wheel speed.
It sounds like what you are doing is exploiting click here biased wheel, which is a different advantage play.
This allows the player to predict where the ball will land based on ball location and past results.
The standard deviation is the square root of the variance, or 15.
The reason for subtracting 0.
Suffice it to say it is an adjustment factor for using a https://clearadultskin.com/best/best-casinos-to-play-poker-in-europe.html function to estimate a discrete function.
Doing a Gaussian approximation, the probability of hitting your number that many times, or more, is 0.
So, there is a this web page chance you found a biased wheel.
However, there is still a 1 in 466 chance it was just good luck.
There is a well-known more info about a freeze-out competition between a Your slothville tv entertaining high roller, Kashiwagi, and Donald Trump, that took place 20 years ago.
Assume that Kashiwagi always bet the maximum on Banker.
What is the probability that Kashiwagi will win?
The math works out more easily if he bet on the Player.
I work out a similar problem in roulette at my mathproblems.
It is much more complicated on the Banker bet, because of the 5% commission.
That would result in the distinct possibility of the player overshooting his goal.
This question was raised and discussed in the forum of my companion site.
Counting the first trial, I show the mean is 8.
So the probability of a repeat within 8 numbers is 100% - 45.
I suspect most people would estimate that that probability of a repeat within 8 numbers would be less than that.
So you would be betting on 8 or fewer, and your friend 9 or more.
Basically, whichever side covers best bet playing roulette median of 8 is likely to win.
This question was raised and discussed in the forum of my companion site.
No, he is not correct.
In all bet calculations, round down.
In other words, always try to reach the goal, with just one bet, if you can, without exceeding the goal.
If there are multiple ways to accomplish this, then go with the one with the greatest probability of winning.
What about other games, you might ask?
According to the Discovery Channel voice-over guy, "Everyone agrees that roulette is the best get rich quick scheme in the casino.
Even limiting ourselves to common games and rules, I find craps to be better.
In particular, betting the don't pass and laying the odds.
This assumes the player may lay 6x odds, regardless of the point which is the case when 3x-4x-5x odds are allowed taking the odds.
This probability of success is 0.
In that case, the probability of success using my Hail Mary strategy is 2.
In both cases, greater than the 2.
In all fairness, the Discovery Channel would have never put the insane rant above on the air and was surely looking for something simple that the masses would understand.
Andy was surely giving them something they wanted to hear.
The basic premise of his advice is that if you want to reach a certain goal, then a hit-and-run strategy is much better than letting the house edge grind you down with multiple bets.
That is definitely true and something I've been preaching for 17 years.
Wizard's "Hail Mary" strategy for craps.
In calculating bets, never bet so much that you overshoot the goal.
Also, never make a bet amount that will cause you to get rounded down.
Otherwise, lay whatever you can.
So, I hope Andy and the Discovery Channel are happy.
I've spent days running simulations to prove them wrong.
This question was raised and discussed on my forum at.
What is the probability of the ball landing in 1, 2, and 3 within 4, 5, 6, 7, 8, or 9 spins in roulette?
The general formula is: Pr Ball lands in 1 + Pr Ball lands in 2 + Pr Ball lands in 3 - Pr Ball lands in 1 and 2 - Pr Ball lands in 1 and 3 - Pr Ball lands in 2 and 3 + Pr Ball lands in 1, 2, and 3.
The following table shows the probability of rolling all three numbers for various number of spins from 3 to 100 best bet playing roulette single- and double-zero roulette.
Spins Single Zero Double Zero 3 0.
If bets on these outcomes paid the usual 35 to 1, then the house edge would be exactly 0%.
The house edge on any best book for learning blackjack on the numbers 1 to 36 would be 5.
Compare that to the 5.
My please click for source in this game would be to bet the zero and double-zero only.
If anyone can confirm or deny these rules and pays, please let me know.
I recorded 7,456 spins in roulette.
The results are as follows.
I suspect the wheel is biased but am not sure if the data is conclusive enough to play it.
Winning Number Occurences 0 204 28 214 9 175 26 177 30 203 11 181 7 223 20 205 32 184 17 222 5 224 22 241 34 194 15 210 3 209 24 176 36 203 13 217 1 217 00 more info 27 173 10 195 25 198 29 217 12 197 8 207 19 163 best bet playing roulette 180 18 201 6 186 21 203 33 171 16 164 4 200 23 191 35 163 14 177 2 194 Total 7456 The following graph shows your results in sequential order on the wheel.
The blue line shows your results.
The red line is the number you need, 207.
A chi-squared test on this distribution comes back with a statistic of 68.
The probability of a result this skewed or more is 1 in 725.
I don't think the chi-squared is the perfect test for this situation because it doesn't consider the ordering of the outcomes, but don't know of a better test.
Some have suggested thebut I don't think that is appropriate.
If there are any other appropriate tests, I'm all ears.
I can say if you had bet the 3-number arc around the number 5, you would have had a 10.
However, if you increased that to a 7-number arc, the advantage drops to 2.
If forced to an answer in plain simple English, I would say the wheel shows evidence, but not proof beyond a reasonable doubt, that the wheel is biased.
However, that bias is probably not enough to significantly and confidently overcome the house edge.
Assuming the casino doesn't switch around the wheels among the tables, I would say that more data should be collected before betting large amounts of money.
I'm sorry this answer is so noncommittal.
This question is raised and discussed in my forum at.
In single-zero roulette, what is the mean and median number of spins required for every number to appear at least once?
Answering the mean is much easier, so we'll start with that.
The median is much more complicated.
To find the exact answer, as opposed to using a random simulation, one needs to use a lot of matrix Algebra.
I've discussed how to solve similar problems in other Ask the Wizard questions, so I won't go through the details again.
One example of a similar question is the one on getting a 6-6 pair in the hole three times in a row, as discussed in.
Suffice it to say that the probability of seeing every number in 145 spins is 0.
Thus, the median is 146.
This question is asked and discussed in my forum at.
If a ball landed in red the last 20 spins in roulette, what is the probability it will land in black the next spin?
I think you're wrong about the previous question.
The odds must overwhelmingly favor black.
That's true, but it doesn't matter.
The fact is the past doesn't matter in games of independent trials like roulette.
I've thought of a way to beat the casinos in roulette!
Start with a small wager on any even-money bet, like red or black.
If it loses, then double the bet on the same thing.
Then keeping doubling until it wins.
The winning outcome has to happen eventually and when it does I'll profit my original wager.
What is your opinion?
Also, please don't tell anybody.
This is probably the most popular of all betting systems, known as the Martingale.
Gamblers have been conceiving of it and using it since time immemorial.
Like all betting systems, not only doesn't it beat the house advantage, it doesn't even dent it.
The reason is the gambler will eventually have a bad losing streak where his bankroll isn't enough to make another double.
In your previous answer, you explained why the Martingale doesn't work.
Then how about the opposite, doubling your bet after each win until a desired target is hit?
This is known as the anti-Martingale and is equally worthless.
The times your bankroll gets grinded down to nothing will outweigh the winnings when you hit your target.
Regardless of what betting system you use, or none at all, the more you play, the more your ratio of money lost to money bet will approach 5.
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