by Marcus du Sautoy
Yesterday, Peter Eastgate left Las Vegas with nearly $10 million after winning the 2008 World Series of Poker. There were more than 6,000 entrants to poker’s premier event but the 22-year-old Dane outlasted the game’s greatest exponents to become its youngest winner.
Card games are often considered to be games of pure luck but poker also requires psychology – the ability to read the temperaments of your opponents – and a good handle on mathematics, which is why time and again you see the same faces finishing in the money in major poker tournaments.
Professionals constantly calculate the probability of winning hands with the cards they hold. So, in the final hand of the tournament, Eastgate had to disguise his excitement. The cards he was holding, the ace of diamonds and five of spades, combined with three cards on the table, gave him a straight (ascending numerical cards 1-2-3-4-5). The only way his opponent, the Russian Ivan Demidov, could beat him was with a higher straight. But the maths told him that of the 990 hands that Demidov could hold, only 12 would win, about a 1 per cent chance. Sure enough, Demidov had only two pairs and Eastgate’s maths had won him the 2008 World Series of Poker bracelet.
Even before you start dealing the cards there is a lot of mathematics that it’s worth being wise to. Hustlers and magicians spend years perfecting something called the perfect shuffle, which allows them to dictate where in the deck cards appear. The deck is split exactly into two equal piles and then the cards are perfectly interweaved like a zipper, alternating one at a time from the right and left hand. It is difficult to perform this trick but, once mastered, it can be put to devastating effect. This is because the person holding the cards knows exactly how the cards are arranged.
So, for example, suppose you and an accomplice want to sting two players in a round of poker. Put four aces on top of the pack. After one perfect shuffle the aces are two cards apart. After another perfect shuffle the aces are four cards apart, perfectly placed for you as the dealer to deal your accomplice all four “bullets”.
Magicians exploit an even more amazing mathematical property of the perfect shuffle. If you do it eight times in a row, although the audience is convinced that the pack must be totally random, the magician knows that the deck has returned to its original arrangement. The perfect shuffle is a bit like rotating an eight-sided coin. Each shuffle is like moving the coin round an eighth of a turn. After eight shuffles, just as the coin has returned to its original position, the deck is just as it was before you started shuffling.
But what if you are shuffling cards for a round of poker at home tonight with your friends. How many times should you shuffle the deck to make sure that the cards are properly scrambled?
Mathematicians have analysed the way most of us shuffle. If you are doing a riffle shuffle (also called a dovetail shuffle, the one favoured by dealers in casinos) and there are L cards in your left hand and R cards in your right then a sensible model is to say that there is an L/(L+R) probability that the card is going to fall from your left hand. After analysing the mathematics of this shuffle, it transpires that you need to shuffle the pack seven times for it to become random. Any less than this and the pack retains information from the previous game.
So, if you have aspirations to be sitting there with the finalists at the 2009 World Series of Poker, just remember, it’s the maths that will make you your millions. Lucky players don’t last.