CHANCE WOULD BE A FINE THING

With an eye to getting rich quick, Richard Askwith asks Britain's leading gambling expert to reveal his secrets

"YOUR BEST strategy here," says Dr John Haigh, casting a dispassionate eye around Peter Pan's Amusement Arcade on the Brighton seafront, "is to tell that boy over there that if he doesn't give you pounds 5, you'll tell the management he's under 18 and not accompanied by an adult. Either that, or keep your money in your pocket."

I hesitate to ignore the advice of a man who has a good claim to be considered Britain's leading expert on gambling, but it seems a shame to stand among all these blaring slot-machines, a short walk from Haigh's Brighton home, without at least trying one. I pop in a pound and within five minutes have quintupled my money. So much for gambling theory.

But Haigh, a cheerful Yorkshireman in his late fifties, isn't going to let a little glitch like that put him off his stride. He's on a roll at the moment, and we both know it. After 33 years at the University of Sussex, where he's Reader in Mathematics and Statistics, he has belatedly risen from the academic ranks this year to become that most envied of creatures: a media don. His recently published book, Taking Chances (OUP, pounds 18.99), has proved an unlikely hit that could well change his life. Never mind that it's essentially a book on mathematics and statistics, littered with sentences such as "The place where the maximum occurs is at the largest value of r such that 1r + 1r+1 + 1r+2 + ... + 1N-1 exceeds 1." It's drawn rave notices in the popular press as well as in more serious journals, and fan mail has been pouring in from all over the country.

The purpose of my visit, which apart from this brief excursion takes place in Haigh's cramped and chaotic office, is to acquire a slice of Haigh's statistical wisdom, with the aim (shared by many of his fans) of converting it into financial gain. I soon realise, though, that this is a little simplistic.

For a start, you can see at a glance that Haigh, while respectable, is not rich. His trousers have seen better days; his anorak looks as if it has seen better decades. He may know how to beat the system, but he doesn't appear to have done so.

In fact, it emerges, Haigh never gambles, despite spending countless hours at casinos and racecourses in pursuit of his research. "I have a few Premium Bonds, but that's it. I'm fundamentally risk-averse."

Is gambling a mug's game, then?

"Not necessarily. Some people make money from gambling. And some people who don't make money from gambling are none the less often acting sensibly. If you're down to your last pound, it's perfectly rational to spend it on a lottery ticket, even though you'll almost certainly lose. But when you're not desperate, as I'm not, then it isn't sensible."

What would be a sensible way for a non- desperate person to make money from gambling?

"Become a bookie."

Anything easier? (As Haigh points out, "Some of the best mathematicians in Britain are bookies.")

"There are some very good gamblers, and they are able to make a profit, and the reason for that is that they have enough knowledge of racing and form to be able to spot value. That is, if a horse is on offer at 5:1, and they can see that its objective chance is 4:1, then in the long run they will make money from betting on such horses."

In Taking Chances, Haigh provides a formula - not, as it happens, of his own invention - by which the serious gambler can calculate how much money, if any, to bet on each horse in any given race. The formula runs to two pages and will strike most readers as too complicated for easy racecourse use.

But the real drawback of this approach to gambling is not so much the mathematics as the immensely time-consuming study of form required to calculate each horse's objective probability of winning. Since most gamblers are hoping to get rich without working hard, this rather defeats the purpose. Hasn't he any easier tips to share?

As it happens, he has many, scattered throughout Taking Chances, although few of them relate to what you or I would think of as proper gambling. Top tips derived from Haigh's well of statistical knowledge include the following:

1 When playing the lottery: you can't influence your chances of winning, but you can increase the likelihood that, if you do win, you'll get a large prize (play only on roll-over or superdraws, incline towards high, even numbers, and avoid obvious sequences like "1, 2, 3, 4, 5, 6" and "7, 14, 21, 28, 35, 42").

1 When playing Monopoly: buy and build on the orange squares.

1 When playing squash: as non-server when the score reaches 8-8, always opt for the longer game rather than sudden death (unless you are on the point of complete physical collapse).

1 In tennis: fast servers on fast courts - for example, Greg Rusedski at Wimbledon - should try to serve aces with their second serves.

1 In football: it's worth getting sent off to prevent a certain goal when the game has been going on for 16 minutes or longer.

1 In the home: insure against major catastrophes, but never take out extended warranties on dishwashers, etc - just pay a notional premium to yourself.

And so it goes on: all interesting but little of it obviously relevant to the crucial issue of getting rich quick - except that, as Haigh points out, "If you don't understand probability, you're unlikely to be a very successful gambler." Yet his statistical knowledge also allows him to offer at least a few tips on beating the bookies and casinos.

"There are general rules," he concedes. "The rate of return on short- priced horses is much better than the rate of return on medium- or long- priced horses, because the odds offered are closer to the true probability of the horse winning."

In the wider field of gambling, Haigh offers a golden rule: "In an unfavourable game bet boldly, in a favourable game bet timidly." Since nearly all serious gambling games are unfavourable - unless you know that your opponent is stupider and/or worse-informed than you - this means that your bets should usually be like underdone T-bone steaks, rare and substantial. Ideally, you should also have a fixed objective, which you should try to achieve with as few bets as possible.

Haigh illustrates this neatly by explaining the right and wrong ways to turn pounds 800 into pounds 900 in a casino. "The timid strategy is to cautiously bet pounds 1 on Red every time. You will be in the casino for a very long time, and your chance of reaching your goal will be under 7 per cent. The rest of the time, you lose the whole pounds 800. The bold strategy is to bet pounds 100 on Red. Either you now have pounds 900, so stop; or you have pounds 700. In the latter case, bet pounds 200 on Red. Either you can stop, or you have pounds 500. If you have pounds 500, you bet pounds 400 on Red. If it wins, stop, if it loses you have pounds 100 which you place on an 8:1 odds bet - for example, a "corner" bet of four numbers. You will then either have zero, or pounds 900. And the overall chance that you will have hit pounds 900 is over 88 per cent."

This is useful stuff, but most gamblers probably lack the arithmetical agility to adapt such calculations to their own specific probabalistic situations. And that suggests that the success of Taking Chances has less to do with the practical advice it contains than with a more subtle sense of the glamour of probability theory. Some of the noblest minds in the history of Western thought have been fascinated by the mathematics of gambling, including the two men usually reckoned to have founded the discipline, Blaise Pascal and Pierre de Fermat. Pascal even applied gambling theory to religion, famously arguing that the rational man should put his faith in Christianity on the grounds that if Christianity was false he would lose little from having believed it, whereas if it was true his loss from not having believed would be infinite. ("If you win, you win everything; if you lose, you lose nothing.")

More recently, probability theory has played an important role in military strategy, in economics, and in biology. John Maynard Smith's famous work on games theory helped to develop the crucial biological concept of "evolutionarily stable strategies" - described at length in Richard Dawkins' The Selfish Gene. Haigh spent much of the Seventies working with Maynard Smith in precisely this field.

Other games theorists - Lennox Figgis, JL Kelly, William Feller, and Edward Thorpe (father of card-counting) - have in recent decades applied games theory to any number of areas of popular concern, from gambling laws to television games. But until Haigh has the discipline had not found a populariser capable of catching the public's imagination.

Perhaps we have been sufficiently softened up by those who have gone before to have lost our fear of the often startlingly counter-intuitive operations of the laws of probability. (Well, would you really bet that, of 22 players and one referee in any football match, two will share a birthday? Or that, on the registration numbers of every 20 cars you pass on the motorway, the last two digits will be identical on at least two? Or that, in a coin-tossing game in which you always call heads, your opponent always calls tails and you keep a tally of your correct calls, the least likely outcome - whatever the length of the game - is that the "lead" will be equally shared?) But Haigh's success also reflects the sheer breadth of the different practical applications he has found for his theoretical knowledge.

No situation is too trivial to benefit from Haigh's theories, whether it's the husband who can't remember if it's his wedding anniversary, or the notional Darren, who can't decide whether to turn up late or early for his date with Zoe. Often, the main intellectual trick is simply to ascribe a numerical value to a non-numerical concept (eg the fury of a woman scorned). But the cumulative effect of seeing so many slices of life through a games theorist's eyes (should James Bond flee to the north or the south? Should Tosca and Scarpia betray one another?) is to cast doubt on the efficacy of less analytical world-views. How can one comfort oneself that one prefers to rely on intuition when faced with so much evidence that intuition is often wrong?

Stephen Jay Gould has said that "Misunderstanding of probability may be the greatest of all impediments to scientific literacy." After a large dose of Haigh, you begin to wonder whether it may not be an impediment to other things as well. Including a fast buck. Surely, I plead, there must be some simple way of converting the wad of notes that's currently burning a hole in my pocket into a substantially larger wad?

"The best way for you to get rich quickly from probability theory," says Haigh, "is to find someone who knows less about it than you do and persuade them to play something like Penney Ante."

Alternatives to Penney Ante - which is described in the box, right - include a variant of "spoof", the coin-showing game that you can win with the aid of some graph paper and "a randomising device such as a stop-watch"; and a version of find-the-lady that involves a card which is black on both sides, a card which is red on both sides, and a card which is red on one side and black on the other. These are fun from an intellectual perspective - in the card game, if a card chosen at random from the three is placed on a table with the up-side showing red, is it true that there's a 50:50 chance that the other side will be red too? No, it's twice as likely to be red as black.

In practice, the drawback is that - although the games theorist Warren Weaver claims to have made good money from betting on this paradox in the Twenties - generally you end up getting beaten up by the sucker you are fleecing. Isn't there some safer strategy that Haigh could recommend, involving a respectable organisation like Ladbrokes?

"Oh yes," says Haigh. "You can give them your money slowly, or you can give it to them quickly."

You don't, I reflect, really need a lifetime's immersion in higher mathematics to reach a conclusion like that, and I beat a disappointed retreat, pausing on my way home to make a brief experimental visit to Ladbrokes.

Bet boldy: that means investing the entire wad at once. Short-priced horses offer a better rate of return than long-priced horses: well, the shortest-priced horse on offer in the next race - the 4pm at Newbury - is Fairy Godmother, at 5/6. It's obvious what Haigh would recommend, and, sure enough, Fairy Godmother romps home comfortably.

If only I hadn't switched my bet to Kittiwake, the second-favourite, at the last minute.

HEADS YOU WIN

Penney-ante is a coin-tossing game invented by games theorist W Penney in 1969. It is also a classic example of how a superior understanding of probability can be turned to financial advantage.

Two players repeatedly toss a coin, making a note of the outcome ("heads" or "tails") in each case. Each player chooses a three-toss sequence of outcomes (eg, heads-tails-heads; or tails-tails-heads, etc) to bet on. The winnner is the player whose sequence appears first in the series of actual tosses.

Ostensibly, each player has an equal chance of winning, with a marginal advantage to the player who chooses first. Opponents willing to play you with this advantage should be in plentiful supply. In fact, the player who chooses second will, with the right strategy, always have a much better chance of winning (the actual probability ranges from 23 to 78).

This may seem absurd. Surely no possible pattern is more likely than another? And if one is, surely the player who chooses first has a better chance of choosing it?

But closer thought shows that you should never play this game unless you are choosing second. All the second player has to do is to make the first player's first two choices his own last two choices. (Thus if I choose heads-heads-heads, you choose tails-heads-heads: unless my sequence comes up immediately, yours will always come first.) If the second player also avoids palindromes (THT, HTH, TTT, HHH), he or she can be pretty confident of raking in the winnings for as long as it takes for the first player to smell a rat.