A group of second-year students asked me to contribute a ‘Real World Stats’ piece to their new psychology publication, MAZE. I reworked a section from of my most popular statistics lectures on probability theory and roulette. Below is the article in full.

Mar 17, 2016 The best red / black system for most people If precognition is not something you want to try, then the best red & black roulette system for you will probably be a basic progression system. Simply if you lose, increase your bet by one unit. If you win, reduce your bet by one unit. European Roulette: Single zero is positioned between black 26 and red 32. Another plus of the European game is en prison rule on even-money bets. When zero comes up, your bet remains or is 'captured' for the next spin. En-prison or surrender rules on even money bets reduces the casino edge further to 1.35%. It does this by combining red/black bets with column bets. If you look at the 3 available columns on a standard roulette table, you'll notice that the first has 6 pockets of each color, but the second and third have a skewed distribution. The second column, illustrated below, has 8 black spaces and 4 red spaces.

Roulette is a straightforward casino game. While the wheel is spinning, a ball is released. This ball eventually ends up stopping in a numbered (1-37) and coloured (red or black) pocket. You bet on the final resting place of the ball by selecting a number, range of numbers or a colour. When betting on colours, if you pick correctly, you double your money. A £20 stake on black would get you £40 back if the ball landed in a black pocket, and nothing back if it landed in a red one.

A few years ago, I received quite a few spam e-mails with the following tip on how to win at roulette.

> So I found a way you can win everytime:
> bet $1 on black if it goes black you win $1
> now again bet $1 on black, if it goes red bet $3 on black, if it goes red again bet $8
> on black, if red again bet $20 on black, red again bet $52 on black (always multiple
> you previous lost bet around 2.5) if now is black you win $52 so you have $104 and you bet:
> $1 + $3 + $8 + $20 + $52 = $84 So you just won $20 :)
> now when you won you start with $1 on blacks again etc etc. its always
> bound to go black eventually (it’s 50/50) so that way you eventually always win.

If you ignore the atrocious spelling and grammar, the basic idea seems to be a good one. In fact, it’s an established betting strategy known as the Martingale system. Under this system, you double losing bets until you win, that way you will always win an amount equivalent to your first stake. If we build a probability tree for a gambler who only bets on black and provide her with a fairly standard outcome, two losses followed by one win, you’ll see how this is meant to work.

Over three bets, she has spent £7, but won £8. Not too shabby. She just needs to do this over and over until she has won an amount she’s happy with. Fool-proof, right?

Not quite. Casinos have stayed in business over centuries for a reason: they know how to work probabilities. One of their standard strategies is to have minimum and maximum stake limits, with a typical range of £10-£1000. These limits expose a huge flaw in our spam-based strategy.

Imagine you’re trying the Martingale strategy and you go on a losing streak. £10, £20, £40, £80, £160, £320 and £640 all go on losing bets and all of a sudden you’re down £1270. Here’s where you come up against the casino’s maximum bet policy. You can’t place a £1280 bet to recoup your losses. But how likely is losing 7 bets in a row?

Not very likely at all, if you’re only trying to win £10. According to the multiplication rule for independent events, the exact odds are (1/2)⁷ which is equal to .0078. Put another way, the probability of this happening is 1 in 128.

But problems arise when you try to make more than £10. To understand the next set of calculations, we need to reverse the probability of losing and think about how likely it is that we will win £10 each time we try. Using the addition rule for mutually exclusive events, we can calculate that the probability of winning £10 is equal to the probability of not losing:

p_win+ p_lose = 1

p_win= 1 – p_lose

p_win= 1 – 1/128 = 127/128 = .9922

We can now work out the probabilities of making various amounts of profit, once again using the multiplication rule:

£20 profit = (127/128)² = .9844
£100 profit = (127/128)¹⁰ = .9246
£200 profit = (127/128)²⁰ = .8548

But here’s the kicker. If you want to double the money you bring to the casino to place these bets, you’re looking at close to a 2 in 3 chance that you will lose everything.

£1270 profit = (127/128)¹²⁷ = .3693

“I’m not greedy!” I hear you cry. “I’d just want to go home with a little more than if I had invested the money and not had any fun at all.” Let’s say you wanted to take home a little more than, 6%, the best savings interest rate you can currently find on moneysupermarket.com (as of when this article was written). How much would you need to win?

£1270 x .06 = £76.20

You would need to win 8 times in a row to go home with a little more than a 6% interest rate. And what are the odds of this happening?

£80 profit = (127/128)⁸ = .9392

Put another way, 15 out of 16 times, you will exceed a savings account interest rate. You will enter the casino with £1270 and leave with £1350. But, 1 in 16 times you will leave the casino with nothing. Not even enough to get the 99 bus back over the Tay. Sadly, this sort of thing is all too common, especially when people are new to gambling and thing they have found a way of beating the system: e.g. http://casinogambling.about.com/od/moneymanagement/a/martingale.htm.

Even if you find a casino with no maximum bet, you need huge financial resources to make it work. It all starts to seem even more hopeless when you factor in something I neglected to mention at the start. Your odds of winning are actually worse than 50%. If the ball lands on 0 the casino takes all the money.

The take-home-message? It’s probably best to ignore financial advice you read in your spam folder.

EDIT (17/1/2014): A link to moneysupermarket.com was removed following receipt of an email from a moneysupermarket.com employee requesting that I comply with their request of ” removing or adding a nofollow attribute to the links to our MoneySuperMarket.com website”. Doing a quick search to find out why reveals https://groups.google.com/forum/#!topic/FleetStreet/HDVHXFwQFdI, which suggests that this is all about SEO optimisation such that “some [links] may look un-natural or paid for in the eyes of Google. This unfortunately means that we have to take down a large number of our links, some of which were genuine and of use/interest to users of the sites on which they were posted”. The suggestion on that last link is that moneysupermarket.com might be doing this because they have previously been penalised for paying for links to their site and are now doing what they can to stop this perception. I don’t know what authority they have to enforce removal of links like this (none I suspect), but I don’t really care enough to kick up a fuss… link removed.

With the exception of a tiny handful of punters, no player can beat the tables consistently without taking advantage of great streaks of luck. Beating roulette with streak bets can be both fun and profitable.

Most Red Or Black In A Row In Roulette Games

Roulette players tend to love their game so much that they adhere to what Nick “The Greek” Dandalos said, “The only thing as good as gambling and winning, is gambling and losing.”

In an effort to find any way possible to supplement or enhance their chances at the roulette table, enthusiasts tend to use betting systems that increase their time at the table without necessarily increasing their overall odds of winning.

The simple fact of roulette is that the house has the advantage and will win one more bet out of every 37 spins than the player. However, roulette betting systems can take advantage of streaks and provide a player with the opportunity to increase their bankrolls in specific situations.

Betting Against a Streak

The Martingale system is one of the best known and easiest to apply betting systems. The player simply doubles their bet each time they lose and waits for a winner. Many roulette players employ this system with an even money bet (odd or even, red or black, first 18 or second 18) after seeing the opposite side hit three times in a row.

The logic that the other side is now due to hit isn’t actually logic at all, it’s the classic gambler’s fallacy that the odds of an event happening are going to even-out quickly by returning a series of events that are the opposite of what has just been observed.

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.

Most Red Or Black In A Row In Roulette Table

So, if you see four straight black spins, does that mean the next spin has to be red (or green) except in one case out of 37? No. The odds are still 48.7% that black is coming back. However, that’s what keeps Martingale players using their system. They will win their small bet many times before they run into a streak that wipes out their current table bankroll.

A player will see six consecutive black spins about once in 75 spins, seven consecutive about once in 154 spins. That’s why Martingale players can win so many times, build up a little bankroll, and have it in their head that the system is a winner. Because of those long odds, a player may win 100, 200, or even 300 straight betting opportunities.

Want to give it a try yourself? Give Lonnie’s roulette simulations a look – click on bet, make the number of spins 100, and see if you get a streak of over 7 on an even-money bet. The results might surprise you!

Dodging the bullet of casino odds is like Russian roulette and you’ve got five chances out of six of winning, and one chance in six that will blow your bankroll out of the water. The problem with the Martingale and many other roulette systems is that you are constantly risking a small fortune to win a single bet. Give it a shot, win a few bets, and consider yourself lucky. Don’t expect it to always work.

Betting With a Streak

Most Red Or Black In A Row In Roulette Random

Instead of betting against a streak, some players have had great success by anticipating a streak of even-money outcomes continuing. With this system, the player simply chooses an even-money bet (odd/even, red/black, 1-18/19-36) and places a single five-unit wager. When it loses it is replaced with a new five unit wager. When it wins, let the winnings ride until it wins five straight times for a payoff of 155 units.

If every bet was a win or a loss, the house would win its 2.7% and that would be it. However, due to several small streaks happening before a streak of five straight appears, the bettor will not lose 19 of every 37 spins. However, it can be a good stretch before a winning streak of five straight appears.

An added twist to this strategy is to pull back a single chip after the first three wins of 5, 10, and 20, so 35 rides on the bet. If this wins, two chips are pulled back so 60 rides. If the bet wins again, either the total of 120 is pulled back, or four chips are pulled back and 100 rides. If the 100 bet wins, the entire 200 is now pulled back and the opposite side of the bet (even – now odd, etc.) is made for a single 5-unit wager.

Most Red Or Black In A Row In Roulette Room

Experiment with your own mixture of these bets to find a happy medium. If you don’t have a wheel at home, try the free online roulette game on the American, European or French roulette pages.

Most Red Or Black In A Row In Roulette Numbers

Barring that, you can always take a deck of cards and make your own set-up of 18 even numbers, 18 odd numbers, and an ace to represent the zero, and shuffle away to try a streak system.