Breaking the Bank in Roulette, with Statistics (or not)

Breaking the Bank in Roulette, with Statistics

Gambling is an interesting intersection of statistics, probability, real life and greed.  That generated the following question on Quora:

As the spindle on a roulette wheel becomes worn over time, does the probability of certain numbers showing up as winners increase?

 

I can recall a story about a gambler in the 19^th century that noticed that the roulette wheels at Monte Carlo were becoming biased by overuse, and supposedly broke the bank by betting on numbers that had become favored in this way.  Eventually the casino figured out that there was a problem, and responded by moving the wheels around the casino overnight, so nobody could be sure about which wheel was which, from day to day.  There is some speculation that the casino colluded in this gambling coup, as the stunt was good publicity.  This was supposed to be the basis of the song “The Man Who Broke the Bank of Monte Carlo”.

There is a related 20^th century version of this tale, whereby some physicists, mathematicians and computer scientists were supposed to have secretly tracked roulette wheels in Vegas in the 1980’s, and used equations of motion from classical physics, in an attempt to “beat the house”.  This is described in the book “The Pleasures of Probability” (Richard Isaac, page 72-73). They claimed to have computers in their shoes (shades of Maxwell Smart) that performed the calculations.  Supposedly, their method worked (at least in theory), but was plagued by technical problems, so they didn’t upset Vegas too badly.

Both of these stories seem rather far-fetched, as it would take a lot of observations to be sure that a number on a roulette wheel was truly coming up much more frequently than chance would predict, especially given the shift in probability needed to break even, let alone make money.

I did my own Monte Carlo simulation (the technique is named after the gambling locale) to simulate games of roulette, just looking at single number bets for simplicity.  A couple of relevant points about roulette are:

• There are 38 numbers on the wheel (1-36, 0, and 00).  So, the probability of the wheel landing on any given number, over the long run, is 1 in 38, which is about 2.63%.
• The return on a correct bet is 36 to 1, which would require a probability of about 2.78% for the bet to be an even money proposition.  In other words, the “bad wheel” would have to move the probability of some particular number winning from 2.63% to 2.78%.

Having a number go from 2.63% hits to 2.78% might not seem all that noticeable to the house (and thus might be ignored), but the player would have to be sure that the wheel truly had become biased by overuse, before it would be a reliable bet.

So, for this technique to be profitable, the gambler would have to watch the wheel, note the numbers that come up, and compute those probabilities.  Once it was clear that a particular number really was coming up more than 2.78% of the time, the gambler could hit that number, confident that the game had turned in the gambler’s favor.

So, how long would a gambler have to watch and record a wheel to be sure that it was a wheel that had “gone bad”, and not a fair wheel showing random variation?  Here are results from some simulations that I did in Excel using the random number function (note that different runs of the simulations could produce different results, but the general trend would remain):

• After 100 spins, the gambler would barely know anything reliable about the wheel.  A perfectly fair wheel could have any particular number getting anywhere from 0% to 7% hits, in 100 spins, just by random chance.
• After 1000 spins, he/she would still be in the dark.  A perfectly fair wheel could have a particular number getting anywhere from 1.5% to 3.6% hits, in 1000 spins. 

• After 10,000 spins, the situation would be little better.  A perfectly fair wheel could have a particular number getting anywhere from 2.2% to 3.0% hits, in 1000 spins.
  
  
• At 100,000 spins, the gambler could finally be reasonably sure that the wheel had turned, and the game could be trusted to be break-even.  With that many simulations, any particular number was hit from 2.53% to 2.78% of the time, in my simulation.

      

• With one million spins, the fair wheel will show all 38 numbers coming up almost the same proportion of times, from 2.61% to 2.67%.

      

So, it looks to me like a gambler would have a difficult time being sure that an apparent “hot number” wasn’t just random variation in a fair wheel.  A bias in an actual bad wheel might become obvious to the gambler much quicker than this, but it would probably be just as noticeable to the house, so it would quickly be shut down.  In fact, I would assume that in this day and age, roulette wheels probably have some kind of counter device installed, to ensure that they don’t develop a noticeable bias.

So, to answer the original question, the probability of certain numbers coming up as winners could increase, via a worn spindle (or other mechanical problem), but it probably wouldn’t be useful for a given gambler to help him/her beat the house.  And, as long as gamblers were ignorant of the bias, it probably wouldn’t affect the casino’s profits, as the gamblers would win a bit more when they happed to bet on the hot number, but lose a bit more when they happened to bet on the others.

Sources:

The Man Who Broke the Bank of Monte Carlo

https://www.dailymail.co.uk/news/article-3724676/The-extraordinary-story-Briton-broke-bank-Monte-Carlo-relentless-casino-winning-streak.html

Physics, Roulette Gambling Coup Attempt:

The Pleasures of Probability, Richard Isaac (Page 72-73, and lots more about roulette and other gambling games)

And here are some other gambling related posts (horseracing).  It’s about my attempt, in younger years, to beat the horses via computer-based statistical analysis:

https://dodecahedronbooks.blogspot.com/2017/08/horse-racing-days-part-1.html

https://dodecahedronbooks.blogspot.com/2017/08/horse-racing-days-part-2.html

https://dodecahedronbooks.blogspot.com/2017/08/horse-racing-days-part-3.html

https://dodecahedronbooks.blogspot.com/2017/08/horse-racing-days-part-4.html

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Now that you have read about gambling systems, here’s a short story about a system that worked, but far too well:

A Dark Horse

Just what might a gambler give up, to go on the winning streak of his life? Even he can't know for sure. Christopher Marlowe's Doctor Faustus legend is given a Damon Runyon spin, in this short story.  For those who aren’t familiar with it, the Faustus legend is about someone who sells his soul to you know who, for fame and fortune.  Things are not nearly so simple for the character in the story, though.

This is a short story of about 6500 words, or about 35 to 45 minutes reading time, for typical readers.

Amazon U.S.: https://www.amazon.com/dp/B01M9BS3Y5

Amazon U.K: https://www.amazon.co.uk/dp/B01M9BS3Y5

Amazon Canada: https://www.amazon.ca/dp/B01M9BS3Y5

Amazon Germany: https://www.amazon.de/dp/B01M9BS3Y5

Amazon Australia: https://www.amazon.com.au/dp/B01M9BS3Y5

Amazon India: https://www.amazon.in/dp/B00OX60XJU

  

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Or how about a nice travel story.

A Drive Across Newfoundland

U.S.: https://www.amazon.com/dp/B07NMR9WM8

U.K.:  https://www.amazon.co.uk/dp/B07NMR9WM8

Germany: https://www.amazon.de/dp/B07NMR9WM8

Japan: https://www.amazon.co.jp/dp/B07NMR9WM8

Canada: https://www.amazon.ca/dp/B07NMR9WM8

Australia: https://www.amazon.com.au/dp/B07NMR9WM8

India: https://www.amazon.in/dp/B07NMR9WM8

Newfoundland, Canada’s most easterly province, is a region that is both fascinating in its unique culture and amazing in its vistas of stark beauty. The weather is often wild, with coastal regions known for steep cliffs and crashing waves (though tranquil beaches exist too). The inland areas are primarily Precambrian shield, dominated by forests, rivers, rock formations, and abundant wildlife. The province also features some of the Earth’s most remarkable geology, notably The Tablelands, where the mantle rocks of the Earth’s interior have been exposed at the surface, permitting one to explore an almost alien landscape, an opportunity available on only a few scattered regions of the planet.

The city of St. John’s is one of Canada’s most unique urban areas, with a population that maintains many old traditions and cultural aspects of the British Isles. That’s true of the rest of the province, as well, where the people are friendly and inclined to chat amiably with visitors. Plus, they talk with amusing accents and party hard, so what’s not to like?

This account focusses on a two-week road trip in October 2007, from St. John’s in the southeast, to L’Anse aux Meadows in the far northwest, the only known Viking settlement in North America. It also features a day hike visit to The Tablelands, a remarkable and majestic geological feature. Even those who don’t normally consider themselves very interested in geology will find themselves awe-struck by these other-worldly landscapes.

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