Go into any casino with normal blackjack rules and you can have a modest advantage without much effort
Basic card counting is simple. Blackjack starts with each player being dealt two cards face up. Face cards count as 10 and the ace as 1 or 11 at the player's discretion. The aim is to have as high a total as possible without "busting" - going over 21. To win, you must achieve a score higher than the dealer's. Cards are dealt from a "shoe" - a box of cards made up of three to six decks. Players can stick with the two cards they are dealt or "hit" and receive an extra card to try to get closer to 21. If the dealer's total is 16 or less, the dealer must hit. At the end of each round used cards are discarded.
The basic idea of card counting is to keep track of those discarded cards to know what's left in the shoe. That's because a shoe rich in high cards will slightly favour you, while a shoe rich in low cards is slightly better for the dealer. With lots of high cards still to be dealt you are more likely to score 20 or 21 with your first two cards, and the dealer is more likely to bust if his initial cards are less than 17. An abundance of low cards benefits the dealer for similar reasons.
If you keep track of which cards have been dealt, you can gauge when the game is swinging in your favour. The simplest way is to start at zero and add or subtract according to the dealt cards. Add 1 when low cards (two to six) appear, subtract 1 when high cards (10 or above) appear, and stay put on seven, eight and nine. Then place your bets accordingly - bet small when your running total is low, and when your total is high, bet big. This method can earn you a positive return of up to 5 per cent on your investment, says Thorp.
After a bit of practice at home, I head off to my nearest casino. Trying to blend in among the rich young things, the shady mafia types and the glamorous cocktail waitresses was one thing; counting cards while trying to remain calm was another. "If they suspect that you're counting cards, they'll ask you to move to a different game or throw you out completely," one of the casino's regulars tells me.
After a few hours I begin to get the hang of it, and eventually walk away with a profit of £12.50 on a total stake of £30. The theory is good, but in practice it's a lot of effort for a small return. It would be a lot easier if I could just win the lottery. How can I improve my chances there?
In it to win it14 January 1995 was an evening that Alex White will never forget. He matched all six numbers on the UK National Lottery, with an estimated jackpot of a massive £16 million. Unfortunately, White (not his real name) only won £122,510 because 132 other people also matched all six numbers and took a share of the jackpot.
There are dozens of books that claim to improve your odds of winning the lottery. None of them works. Every combination of numbers has the same odds of winning as any other - 1 in 13,983,816 in the case of the UK "Lotto" game. But, as White's story shows, the fact that you could have to share the jackpot suggests a way to maximise any winnings. Your chances of success may be tiny, but if you win with numbers nobody else has chosen, you win big.
So how do you choose a combination unique to you? You won't find the answer at the National Lottery headquarters - they don't give out any information about the numbers people choose. That didn't stop
Simon Cox, a mathematician at the University of Southampton, UK from trying. Ten years ago, Cox worked out UK lottery players' favourite figures by analysing data from 113 lottery draws. He compared the winning numbers with how many people had matched four, five or six of them, and thereby inferred which numbers are most popular.
And what were the magic numbers? Seven was the favourite, chosen 25 per cent more often than the least popular number, 46. Numbers 14 and 18 were also popular, while 44 and 45 were among the least favourite. The most noticeable preference was for numbers up to 31. "They call this the birthday effect," says Cox. "A lot of people use their date of birth."
Several other patterns emerged. The most popular numbers are clustered around the centre of the form people fill in to make their selection, suggesting that players are influenced by its layout. Similarly, thousands of players appear to just draw a diagonal line through a group of numbers on the form. There is also a clear dislike of consecutive numbers. "People refrain from choosing numbers next to each other, even though getting 1, 2, 3, 4, 5, 6 is as likely as any other combination," says Cox. Numerous studies on the US, Swiss and Canadian lotteries have produced similar findings.
To test the idea that picking unpopular numbers can maximise your winnings, Cox simulated a virtual syndicate that bought 75,000 tickets each week, choosing its numbers at random. Using the real results of the first 224 UK lottery draws, he calculated that his syndicate would have won a total of £7.5 million - on an outlay of £16.8 million. If his syndicate had stuck to unpopular numbers, however, it would have more than doubled its winnings (
The Statistician, vol 47, p 629).
So the strategy is clear: go for numbers above 31, and pick ones that are clumped together or situated around the edges of the form. Then if you match all six numbers, you won't have to share with dozens of others.The lotto strategy is clear: go for numbers above 31, and pick ones that are situated around the edges of the ticket. Then if you do win, you'll win big
Unfortunately, probability also predicts that you won't match all six numbers until the 28th century. I bought a ticket using some of Cox's least popular numbers: 26, 34, 44, 46, 47 and 49. Not one of them came up. So I headed for the bookmaker.
Although it would be nearly impossible to beat a seasoned bookie at his own game, play two or three bookies against each other and you can come up a winner. So claims
John Barrow, professor of mathematics at the University of Cambridge, in his book
100 Things You Never Knew You Never Knew. Barrow explains how to hedge your cash around different bookies to ensure that whatever the outcome of the race, you make a profit.
Although each bookie will stack their own odds in their favour, thus ensuring that no punter can place bets on all the runners in a race and guarantee a profit, that doesn't mean their odds will necessarily agree with those of a different bookie, says Barrow. And this is where gamblers can seize their chance.
Let's say, for example, you want to bet on one of the highlights of the British sporting calendar, the annual university boat race between old rivals Oxford and Cambridge. One bookie is offering 3 to 1 on Cambridge to win and 1 to 4 on Oxford. But a second bookie disagrees and has Cambridge evens (1 to 1) and Oxford at 1 to 2.
Each bookie has looked after his own back, ensuring that it is impossible for you to bet on both Oxford and Cambridge with him and make a profit regardless of the result. However, if you spread your bets between the two bookies, it is possible to guarantee success (see diagram, for details). Having done the calculations, you place £37.50 on Cambridge with bookie 1 and £100 on Oxford with bookie 2. Whatever the result you make a profit of £12.50.
Simple enough in theory, but is it a realistic situation? Yes, says Barrow. "It's very possible. Bookies don't always agree with each other."
Guaranteeing a win this way is known as "arbitrage", but opportunities to do it are rare and fleeting. "You are more likely to be able to place this kind of bet when there are the fewest possible runners in a race, therefore it is easier to do it at the dogs, where there are six in each race, than at the horses where there are many more," says Barrow.
Even so, the mathematics is relatively simple, so I decided to try it out online. The beauty of online betting is that you can easily find a range of bookies all offering slightly different odds on the same race. "There are certainly opportunities on a daily basis," says Tony Calvin of online bookie Betfair. "It's not necessarily risk-free because you might not be able to get the bet you want exactly when you need it, but there are certainly people who make a living out of arbitrage."