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Interesting math behind this industry, or why you really should consider matched - betting, instead regular, risky one.

SCOTTISH FINANCIAL wizard John Law is remembered, among other things, for setting up France's first central bank in the early 18th century. Like a lot of people who work in that business, he was fond of a bet.

One of his favourite tricks was to offer people odds of 10,000-1 that they could not throw six consecutive sixes with a die. It was bad value. The real odds are over 46,000-1. Law was never likely to pay out.

It is probably one of the first recorded examples of a house edge. This is the difference between the real odds and those offered to the punter. For example, the odds against any single number coming up on a roulette wheel are 36-1, but casinos pay you 35-1, giving them a 2.7 per cent safety net.

All businesses that make money from gambling focus on balancing risk with offering their customers attractive odds. Casinos have a house edge. Bookmakers have an over-round, designed to give them a margin that will guarantee a profit but ensure that their punters win a share of money.

"We actually want people to have that winning experience," Paddy Power, head of communications with the bookmaker's chain of the same name, says. "If someone comes in and keeps losing, they'll just stop coming back."

The over-round works like this: a bookie offers two punters bets on the toss of a coin. The odds on either outcome are even money, 50 per cent plus 50 per cent, or 100 per cent. This is a pure betting market, as the odds fairly reflect the probability of either outcome. But the bookie cannot make money at those odds, so he will offer something like 10-11 on both results.

This means that both punters have to stake €11 in order to win €10. So the bookmaker takes in €22, and pays out €21 (the €11 stake plus €10 in winnings) to the winner, and keeps €1 for himself.

The bookie's odds come to 104.8 per cent, that is, 52.4 per cent plus 52.4 per cent, giving him a margin of 2.4 per cent on each result. The 4.8 per cent is the over-round.

The house edge and over-round ensure that the return is a few per cent below the real risk, which means that as all eventualities play out, punters will ultimately lose more than they win.

Power's colleague, Dermot Golden, the firm's head of risk, says there is more to it than offering 10-11 on a 50/50 chance. He points out that customers know when they are not getting decent odds. "It's a very competitive market and they will go somewhere else if they don't believe that they are getting value," he says.

Golden says bookmakers have to offer proportionately more generous odds on the fancied horses that people want to back, than on the outsiders that go off at big prices. This means that even if a bookmaker's margin on a race averages 2 per cent a runner, it is likely to be 1.5 per cent on the popular runners.

Just a few horses in every race generate most of the action. When the Aidan O'Brien-trained Yeats won his historic third Ascot Gold Cup last Thursday, he beat 10 others. There was £182,550 in bets of £500 or more wagered at the course. Yeats, the favourite, attracted £141,500, the second favourite carried £40,000 and the balance went on two others.

Despite such exposure to favourites, Boylesports chief executive Daniel O'Mahony says that in some situations being generous can work in the bookmaker's favour. The firm tends to give better odds than most others on certain football bets. But it does so for a reason. "What we've found is that people start to see value on both sides, and it tends to spread the risk more evenly," he explains.

There is a perception that bookmakers spread risk by "laying off", that is, betting part of their liability on a particular result with a rival at the same or better odds. However, O'Mahony says that firms only do this rarely. "Laying off will cost you money over time; our response in that situation is to keep cutting our odds, that's how we manage the risk," he says.

Firms such as Boyles and Paddy Power employ odds compilers who work out the initial prices that each will offer on the next day's sporting events. They then advertise these each morning. If a horse or particular result begins to attract a lot of money, they will shorten the odds accordingly.

All this underpins consistently profitable businesses. Paddy Power, Ireland's biggest chain, made €72 million last year from its betting operations, a big increase on the €45 million it earned in 2006. Britain's biggest player, Ladbrokes, which has a large Irish business, made £420 million in 2007, up from £260 million the previous year. Paddy Power's clients bet €2 billion with it in 2007, and lost €279 million, or 14 per cent. The average bet placed by its customers during the year was just over €20, so each bet cost them €2.80.

So can you beat the bookie? Gamblers say the best chance of doing so is by focusing on "value". Dr Patrick Murphy, a statistician (not a gambler) with UCD's school of mathematical sciences, has a working definition of this: value is where the payback exceeds the real odds.

This seems a difficult concept to apply to a horse race, but if you had €1 on every horse trained by Aidan O'Brien and ridden by Johnny Murtagh in group one races at Ascot last week, you would have made €8.25.

This is not an accident. O'Brien trains mainly for Coolmore stud, which breeds and buys top quality horses with the intention of turning them into stallions. To do that, they have win top-level races at high-profile meetings.

It is five years since the bookies last lost at the Cheltenham festival. But every year, horses that have won previously at the course outperform expectations at the festival. Three of this year's four championship race winners had won there previously. That is not an accident either. Cheltenham is a difficult track that does not suit every horse, but brings out the best in others.

Such "rules" are not hard and fast, but another mathematician, Edward O Thorp, developed one that is. A teacher at the University of California, Los Angeles, in the 1960s, he discovered something about the casino card game, blackjack, that reversed gambling's laws. The house edge in blackjack is 5 per cent, but Thorp found that when the deck from which the casino's croupier is dealing contains a large number of high-value cards, the odds swing in the players' favour.

He came up with a system of tracking or "counting" cards, tried it, found that it worked and wrote a book, Beat the Dealer , detailing it. The casinos laughed at him, but banned him when it became clear that he was right.

Thorp moved on to the biggest casino of all, global financial markets, where he started one of the first hedge funds, and made a killing on the break up and reflotation of ATT.

John Law was not so lucky. He took over France's financial system in 1718. To raise money, he started the Mississippi Trading Company, and sold shares, inflating a huge investment bubble. It collapsed, taking the economy with it. Law, who had a murder conviction hanging over him in London, fled for the gambling dens of Europe. History does not record, but the odds are that he never had to pay out.

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