Category Archives: Gambling

Here’s the best way to shuffle a pack of cards – with a little help from some maths

Graham Kendall, University of Nottingham

Shuffling a pack of cards isn’t as easy as you think, not if you want to truly randomise the cards. Most people will give a pack a few shuffles with the overhand or riffle methods (where the pack is split and the two halves are interweaved). But research has shown this isn’t enough to produce a sufficiently random order to make sure the card game being played is completely fair and to prevent people cheating. The Conversation

As I wrote in a recent article about card counting, not having an effective shuffling mechanism can be a serious problem for casinos:

Players have used shuffle tracking, where blocks of cards are tracked so that you have some idea when they will appear. If you are given the option to cut the pack, you try and cut the pack near where you think the block of cards you are tracking is so that you can bet accordingly. A variant on this is to track aces as, if you know when one is likely to appear, you have a distinct advantage over the casino.

Card Counting and Shuffle Tracking in Blackjack.

So how can you make sure your cards are well and truly shuffled?

To work out how many ways there are of arranging a standard 52-card deck, we multiply 52 by all the numbers that come before it (52 x 51 x 50 … 3 x 2 x 1). This is referred to as “52 factorial” and is usually written as “52!” by mathematicians. The answer is so big it’s easier to write it using scientific notation as 8.0658175e+67, which means it’s a number beginning with 8, followed by 67 more digits.

To put this into some sort of context, if you dealt one million hands of cards every second, it would take you 20 sexdecillion, or 20,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, years to deal the same number of hands as there are ways to arrange a deck of cards.

You would think that it would be easy to get a random order from that many permutations. In fact, every arrangement is, in a sense, random. Even one where the cards are ordered by suit and then rank could be considered random. It is only the interpretation we put on this order that would make most people not consider it random. This is the same as the idea that the lottery is less likely to throw up the numbers one to six, whereas in reality this combination is just as probable as any other.

In theory, you could shuffle a deck so that the cards emerged in number order (all the aces, followed by all the twos, followed by all the threes and so on), with each set of numbers in the same suit order (say spades, hearts, diamonds and clubs). Most people would not consider this random, but it is just as likely to appear as any other specific arrangement of cards (very unlikely). This is an extreme example but you could come up with an arrangement that would be seen as random when playing bridge because it offered the players no advantage, but wouldn’t be random for poker because it produced consistently strong hands.

But what would a casino consider random? Mathematicians have developed several ways of measuring how random something is. Variation distance and separation distance are two measures calculated by mathematical formulas. They have a value of 1 for a deck of cards in perfect order (sorted by numbers and suits) and lower values for more mixed arrangements. When the values are less than 0.5, the deck is considered randomly shuffled. More simply, if you can guess too many cards in a shuffled deck, then the deck is not well shuffled.

The Best (and Worst) Ways to Shuffle Cards – Numberphile.

Persi Diaconis is a mathematician who has been studying card shuffling for over 25 years. Together with and Dave Bayer, he worked out that to produce a mathematically random pack, you need to use a riffle shuffle seven times if you’re using the variation distance measure, or 11 times using the separation distance. The overhand shuffle, by comparison, requires 10,000 shuffles to achieve randomness.

“The usual shuffling produces a card order that is far from random,” Diaconis has said. “Most people shuffle cards three or four times. Five times is considered excessive”.

But five is still lower than the number required for an effective shuffle. Even dealers in casinos rarely shuffle the required seven times. The situation is worse when more than one deck is used, as is the case in blackjack. If you are shuffling two decks, you should shuffle nine times and for six decks you need to shuffle twelve times.

Shuffle like a casino dealer.

Many casinos now use automatic shuffling machines. This not only speeds up the games but also means that shuffles can be more random, as the machines can shuffle for longer than the dealers. These shuffling machines also stop issues such as card counting and card tracking.

But even these machines are not enough. In another study, Diaconis and his colleagues were asked by a casino to look at a new design of a card shuffling machine that the casino had built. The researchers found that the machine was not sufficiently random, as they simply did not shuffle enough times. But using the machine twice would resolve the problem.

So next time you’re at a casino, take a look at how many times the dealers shuffle. The cards may not be as random as you think they are, which could be to your advantage.

Graham Kendall, Professor of Computer Science and Provost/CEO/PVC, University of Nottingham

This article was originally published on The Conversation. Read the original article.

Can maths help you win at roulette?

Graham Kendall, University of Nottingham

Albert Einstein supposedly once said: “No one can win at roulette unless he steals money from the table while the croupier isn’t looking.” The Conversation

Although I wouldn’t normally question Einstein, this statement isn’t true. In fact, you can use Einstein’s specialist subject, physics, to help you win. Or you can find a biased wheel that makes some numbers more likely to come up.

What Einstein actually meant was that there is no mathematical trick that can help you win at roulette. Each spin is an independent trial and, in the long run, the casino will win. This is different to a game such as Blackjack where the probabilities change as cards are dealt.

But some believe that it is possible to exploit the way the roulette wheel, and the betting cloth, is laid out to give themselves an advantage. The idea is that you can make bets on the layout in a way that you are guaranteed to win. But is this really possible?

Roulette wheel layout

Like a dartboard, the layout of a roulette wheel did not come about by accident. It was carefully planned and exhibits certain properties. In fact, there are two different layouts. An American wheel and a European wheel. The two layouts are shown below.

American roulette wheel layout.
Wikimedia Commons

European Roulette Wheel Layout.
Wikimedia Commons

Notice that the American wheel has two zeroes. This is important as it doubles the advantage for the casino. On a European wheel you would expect to lose, in the long run, 2.7% of any money you bet with. On an American wheel you can expect to lose 5.26% (if you are interested in the mathematics of roulette, the video at the end will show you how these odds are calculated).

The numbers are arranged in a different order on each wheel but there are some similarities in the patterns. On both wheels, the red and black numbers alternate around the wheel, although if you removed the zeroes, the American wheel would have consecutive reds and blacks. The wheels are also structured so that the low numbers (1-18) and the high numbers (19-36) should alternate as much as possible.

On a European wheel, this is only violated where the 5 sits next to the 10 (both low numbers). On the American wheel, there are many examples where this rule is violated. It is for this reason that the American wheel is considered not as balanced as the European wheel. Both wheels also try to distribute odd and even numbers as evenly as possible. But again there are a number of violations of this rule on both wheels.

On the European wheel there are two other interesting symmetries. First, all the low red numbers and black high numbers are on one side of the zero, and the high red numbers and low black numbers are on the other side. Second, the sequence 29-7-28-12-35-3-26-0-32 contains no numbers between 13 and 24 (the second dozen). You can place a bet on the whole of the second dozen, with odds of 2-1.

European roulette layout.
Wikipedia Commons

So, can we beat the maths?

A simple search on Google will return many (possibly millions) of systems for playing (and supposedly winning) roulette. Some easy, some complicated, some well described, some not so.

A system should really be a combination of a playing strategy and a money management strategy. Perhaps the best known money management strategy is the Martingale system. This system is guaranteed to win money as long as you have enough of a bankroll to double your bet after every loss and you do not hit the table limit, which you will quickly do so. The Martingale system is probably the quickest way to bankruptcy known to man.

Whatever betting strategy, and money management strategy, you choose, they all suffer from the same fate. Assuming that each number on the wheel has the same probability of being selected – meaning the wheel is not biased – the maths means the casino will always win. The system may look good, and may work in the short term, but when one of the numbers comes up that you have not bet on you will lose and the casino will move towards its win expectation (2.7% or 5.26%).

Some systems involve betting on many numbers, perhaps 20. In this case, you will win quite often as you are covering more than half of the numbers. But when one of the numbers does not turn up (and it will almost half the time) you lose all of the 20 bets you have made. This will often wipe out any wins to date.

Any system, so far devised, can be analysed to show that there is a win expectation for the casino. The following video shows the maths.

The mathematics of roulette.

You might as well place a single chip on the same number every time and hope that it appears more than it should during the short time that you are playing.

We can dress up the layout of the wheel, the layout of the betting cloth, our number selection and our money management system however we like, but the maths is always there, quietly working against us. You might as well just have fun, pick random numbers and trust to Lady Luck. Either that, or do as Einstein suggested and steal chips (not that we’d recommend it).

Graham Kendall, Professor of Computer Science and Provost/CEO/PVC, University of Nottingham

This article was originally published on The Conversation. Read the original article.

How Isaac Newton could help you beat the casino at roulette

Graham Kendall, University of Nottingham

Imagine walking into a casino with a computer strapped to your chest. Solenoid electromagnets thump against your body telling you where to place your bet on the roulette table. Suddenly, you start getting electric shocks. You rush to the toilet to undertake emergency repairs, hoping that the casino staff do not realise what is happening. The Conversation

In the late seventies, graduate student Doyne Farmer and colleagues did just that – with purpose-built computers that could predict where a roulette ball would land. The project, described in the book The Newtonian Casino (published as The Eudaemonic Pie in the US), was, however, difficult and fraught with technical problems. The team never really found a reliable way of doing it. But decades later, is it any closer to becoming a reality?

In a game of roulette, the croupier spins a wheel in one direction and a ball in the other direction. Players then place bets on where the ball will land by choosing either a single number, a range of numbers, the colours red or black or odd or even numbers.

Our understanding of the physics behind the movement of the ball and wheel is pretty solid – governed by Newton’s laws of motion. As the ball slows, gravity takes hold and it falls into one of the numbered compartments. It is predictable when the ball will leave the rim. However once it does, the route it takes to a numbered slot is less so. This is because the ball bounces around as it strikes various obstacles.

Every roulette wheel is slightly different. Atmospheric conditions continually change and the wheel itself has features that encourage randomness – such as the size of the frets between the numbers and the diamond-shaped obstacles that intercept the ball as it falls down to the wheel. This means that you cannot predict the exact number where the ball will land. But you only need to know which area of the wheel the ball will land and you can gain a massive advantage over the casino – more than 40%. This is a huge swing from the 5.26% margin that US casinos have over players – often referred to as the house edge. In Europe it is only 2.7%, as the wheel has only one zero (a US wheel has two zeroes).

Sweaty experiments

When Farmer and his team entered the casino for the first time, two people were wearing computers. One had a computer built into his shoes, with the task of inputting data by tapping switches under the toes. This computer performed two main functions. One was to adjust parameters for each wheel before a game, such as the rate at which the ball and wheel slowed down, and the velocity of the ball when it fell off the track. They also had to determine whether the wheel exhibited any tilt.

The second job was during live play. The player with the shoe computer tapped the toe switches each time a certain point (typically the double zero) on the wheel passed by and also when the ball passed by. Using this information, the program could calculate the speed of both the wheel and the ball – thus knowing when the ball would start to fall. Knowing the relative positions of the ball and the wheel meant that a prediction could be made about where the ball would finally land. The computer then had to transmit the prediction to the person wearing the second computer. This was achieved by weak radio signals.

Shoe computer. The Eudaemonic Pie display at the Heinz Nixdorf Museum., CC BY-SA

The second computer, strapped to someone else, received the radio signals and conveyed this information to the player by the solenoid electromagnets that thumped that player’s stomach. A code had been developed which relayed the predicted number, with the player placing bets on that number and several numbers either side to account for the randomness. In order that the casinos could not easily see what they were doing, the team altered their betting patterns slightly. For example, not betting on all the consecutive numbers.

However this never gave them the 40% advantage observed in the lab – mainly due to technological problems such as short circuits caused by sweating, wires becoming loose and lost radio connections.

It took several years for the team (which now comprised about 20 people who’d worked on the project in varying degrees) to develop an improved computer system. Both computers were now in custom-built shoes. This could protect the operator from being electrocuted but would also make it harder for the casino to detect. The other innovation was that the computers were set in resin blocks, with only the toe-operated switches and the solenoids that now drummed against the feet, being visible. This was to try and combat the problems such as loose wires and sweating.

Binion’s casino.
Ken Lund/Flickr, CC BY-SA

They then entered Binion’s casino in Las Vegas ready for an all-out assault. Once the parameters had been set, the first prediction was to bet in the third octant – which included the numbers 1, 13, 24 and 36. The ball landed in 13 and the team got paid off at 35-1. The years of work looked promising, but the solenoids eventually started to act randomly, so the accurate predictions from one computer were not being transmitted to the other. The team suspected it was due to the electronic noise present in casinos. Eventually they had no choice but to abandon the idea.

Would it work today?

The main issue in the late seventies and early eighties was that the team had to build their own computers from scratch, literally – they had to design the computer, buy all the components and get busy with a soldering iron. These days, the computers are readily available, as the following video shows.

Technology has evolved. These days, all the required processing power could be fitted into a single unit. You could imagine a system based on a mobile phone where the camera videos the ball and the wheel and image processing software extracts the relevant data so that the prediction software can calculate the final position of the ball.

But certain challenges still remain. If several people are involved, which is the best way to avoid detection, how can you work as a team and pass data? Perhaps the use of free wifi in many casinos could be a solution? Another problem is how to best hide the fact that you are trying to use an electronic device to predict where the ball will land, when you need to input data and receive the prediction. Here, suitably connected glasses may be one get around, used in tandem with toe-operated switches.

The hardest challenge, however, is the casino itself. They are certainly unlikely to simply let you have a camera pointed at the roulette wheel, especially if you are winning. If they did, they would be likely to ask you to leave and as it is often illegal to use such devices. But with a little creativity it may not be long before scientists prove they are able to outsmart casinos.

Graham Kendall, Professor of Operations Research and Vice-Provost, University of Nottingham

This article was originally published on The Conversation. Read the original article.

How to beat the casino – legally

Graham Kendall, University of Nottingham

If there’s one thing everybody knows about gambling it’s that the house always wins. And while it is true that casinos always make a profit, there are a number of ways to cheat the system – some of which are actually perfectly legal.

Half a century ago, mathematician Edward Thorp published a groundbreaking book outlining how a player could use “card counting” to get an advantage in the game Blackjack by keeping track of the cards left in a deck. Ever since, casinos have been trying to eradicate card counting while card counters are getting increasingly skilled at not getting caught. So is it possible to outplay casinos today? And what will it be like in the future?

Winning Blackjack Hand.
Wikipedia Commons

Casinos are businesses and operate by building in a margin – often referred to as the house edge. If you play roulette and bet on a single number you will be paid at odds of 35-1 when the true odds are 36-1 in Europe and 37-1 in the US. The fact that you are receiving less than the true odds is the house edge and explains why casinos make money in the long term. Of course, some people have to win, otherwise casinos would cease to exist.

Advantage players

What casinos don’t like are “advantage players” – people seeking to have an edge over the house. Sometimes this involves cheating and/or illegal activities ranging from past posting (making a bet after the time when no more bets are to be taken) to collaborating at the poker table and using a computer to help make decisions.

Card counting, however, is legal. In Blackjack, the aim of the player is to achieve a hand of cards whose points add up nearer to 21 than the dealer’s hand, but without exceeding 21. Many hands are played from the same deck of cards, so what happens in one hand will influence what happens in future hands. As an example, if a ten has been played from the pack then it cannot appear in the next hand. This is different from other games, such as roulette, where the outcome of one spin has no effect on the next spin.

Professor Thorp and his contribution to card counting.

Card counting is based on the fact that a large proportion of high cards (such as tens, jacks, queens and kings, which are all worth ten points) left in the unplayed deck statistically improves the player’s chances. This is because a player can decide not to draw a new card to a hand such as 16, but the casino is forced to, as it follows strict rules. If there are a high proportion of high cards left in the unplayed deck of cards, the dealer has more chance of busting (going over 21). This can be combined with “basic strategy” – developed from computer simulations of millions of blackjack hands – which tells the player the best action to take for each possible card combination.

Combining card counting and basic strategy can help a player convert the (long term) house edge from 2.7%, in favour of the casino, to about a 1% advantage to the player. Of course, once you have this advantage you can increase your bet.

To give a simple example, if you were playing basic strategy and were dealt a ten and a six, and the dealer had a three showing (one of the dealers cards is visible to the player), you would stand (not take another card) as you hope that the dealer would draw a ten and bust. If you were card counting, and you knew that more low cards had been played, you might decide to increase your stake at this point.

Evolving battle

Casinos have introduced a number of measures to deter card counting. These include spotting those doing it and simply banning them from playing, or even from entering the casino. Another approach is to increase the number of decks from one to (typically) six, or even eight. Some casinos also shuffle the cards after only about 75% have been played or shuffle them constantly using automatic shufflers.

You might wonder why casinos don’t simply withdraw blackjack. Well, it remains a popular game, and one that is still profitable. There are also many would-be card counters who are not actually that good at it, and they provide income to the casinos.

Many blackjack players have fought back against such measures, arguing that casinos should allow gamblers to use skill when playing the game. As a card counter operating on their own is relatively easy to spot (intense concentration, increasing bets and so on), a team of students from MIT showed it could successfully be done in teams. The idea is that somebody else counts the cards – they may not even be sitting at the table. When the count reaches an agreed value, they signal to another player, who joins the table to start betting. This is a lot more difficult to detect but casinos may stop players joining the game until after a shuffle to combat such a strategy.

Breaking Vegas: the true story of The MIT blackjack team.

Other players have used shuffle tracking, where blocks of cards are tracked so that you have some idea when they will appear. If you are given the option to cut the pack, you try and cut the pack near where you think the block of cards you are tracking is so that you can bet accordingly. A variant on this is to track aces as, if you know when one is likely to appear, you have a distinct advantage over the casino.

It’s been 50 years since Thorp’s book, and it is unlikely that the war of wills between blackjack players and casinos will end any time soon. Some of our work has investigated how artificial neural networks (simple models of the human brain) could help evolve blackjack strategies. This was done by playing thousands of blackjack hands and the computer learning what to do in any given situation, getting better each time. There is a lot of scope to see if automated computer programs could learn even more sophisticated strategies.

Graham Kendall, Professor of Operations Research and Vice-Provost, University of Nottingham

This article was originally published on The Conversation. Read the original article.