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The gambler who cured an archbishop, wounded a politician and counted the odds

The Gambling Scholar's ideas resonate to this day

Nat Lofthouse is honoured with a statue outside Bolton's Macron stadium
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Gerolamo Cardano was not the first man to wonder what are the odds, but he might have been the first to work them out and write them down. Cardano left a manuscript that was published in 1663 – 87 years after he died – as The Book on Games of Chance.

Some historians of maths mention it in passing. Others ignore it. Some of the sums in it are wrong, which is why it has been neglected. But it should not have been. Because the mistakes are not really mistakes.

Cardano in some places gave the wrong answers to the questions he had asked, but they were the right answers to other important questions that he had not even thought of. Without realising, Cardano had discovered probability and expectation, the bases of odds and spreads.

If you have a bet today, with anyone from Sky Bet to Sporting Index, you will be challenging prices that were compiled with ideas that can be traced back to maths written down by Cardano.

Cardano lived between 1501 and 1576 in Italy. He became an eminent doctor. In 1552 he was persuaded to travel all the way to Britain to treat the Archbishop of Scotland. Cardano eased the archbishop’s asthma.

Cardano also wrote books that were published while he was alive. Oystein Ore, a Yale University professor, released in 1953 a biography of Cardano called The Gambling Scholar. Ore said some experts on William Shakespeare’s plays believe the speech in Hamlet that begins “To be or not to be…” was based on passages in Cardano’s book called Wisdom and Consolation.


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Cardano gambled from his teenage years until old age. As a student he needed to win money to buy food. That was pressure. Cardano played cards, dice and something like backgammon. His skill at cards seems to have come from what we call card counting. In The Book on Games of Chance he wrote: “By careful attention I brought it about that I was always mindful of all the cards which I had discarded”.

Cardano believed that only those who have practiced should preach. “So it was right for Hannibal to make fun of the philosopher who had never seen a battle line and was discoursing on war.” Hannibal was the general who came to Europe from north Africa and crossed the Alps with elephants.

Cardano got into some fights when gambling. In Venice he stabbed a senator he accused of cheating. That was dangerous.

Cardano wanted to know the odds for the card, dice and board games he played. He found two ways of calculating chances. The first gave him the right probabilities. The second gave him a different set of numbers, which he realised eventually were the wrong answers for probabilities. What he never realised was that they were the right answers for expectations.

Others might have discovered these things earlier, but if they did either they did not write them down or their writing was lost.


The very different prospects of success and failure

The pain of losing is more intense than the pleasure of winning. That is one of the claims of prospect theory. It was developed by psychologists Daniel Kahneman and Amos Tversky. They unveiled it in 1979 in a paper called Prospect Theory: An Analysis of Decision Under Risk.

Kahneman wrote later: “The reason you like the idea of gaining $100 and dislike the idea of losing $100 is not that these amounts change your wealth. You just like winning and dislike losing – and you almost certainly dislike losing more than you like winning… When directly compared or weighted against each other, losses loom larger than gains.”

Many gamblers would agree. Most good observations had occurred to other people before. This one had been written down more than 400 years earlier by Gerolamo Cardano. He made the same point in The Book on Games of Chance, though he explained it in a different way, using an example that must have come to him watching the people he played against.

The loser is left with his loss, Cardano said, but the winner may not keep what he has won. He is likely to become bolder as his winnings pile up until eventually he overreaches himself and loses all.

Cardano wrote: “If you should lose a large amount, then certainly the gain to the victor is not so great as the loss to the loser; for usually he gives away a great deal; the man is wasteful and his time is lost; and when the die once falls unluckily for him, his loss may be greater than the net gain of many wins.”


An old footballer with some modern insights

Nat Lofthouse played centre-forward for Bolton and England after the second world war. When he had retired he was interviewed by Arthur Hopcraft for a book published in 1968 as The Football Man. It is an illuminating book by a perceptive writer who had spoken to many other players and managers.

Lofthouse made three points that have been made before on this page: that sometimes boy footballers stand out because they are big for their age; that sometimes we have no idea how or why things happen (they are subject to the laws of chance that Gerolamo Cardano explained and there is no other explanation for why we get this rather than that, and now rather than then); and whenever possible attacks should end with an attempt on goal.

Lofthouse played football at school. The games master made him centre-forward. “There was no reason for it then. It was just that I was always a big lad.”

Today there is a statue of Lofthouse outside Bolton’s ground, but when he started playing in what we now call the Premier League he was neither good nor popular. Hopcraft wrote: “He struggled through a long period of derision from Bolton’s supporters and colder criticism from the newspapers. Both kinds of attention hurt.”

Then suddenly something clicked. Lofthouse said: “You just don’t know how it comes… One day you go up for a ball and, bang, it’s in the net. You’ve been doing the same thing for months and getting nothing for it. That’s how it was. No one was more surprised than me.”

Lofthouse scored goals by heading crosses. When he graduated to international football he played between two of England’s best-ever wingers, Stanley Matthews and Tom Finney, and he noticed the difference. “With Matthews and Finney you knew that if you gave them the ball they’d either beat a man and get a cross over to you or they’d manage somehow to give you the ball back. They’d always do something. They wouldn’t just lose the thing.”


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Cardano gave the wrong answers to the questions he had asked, but they were the right answers to important questions that he had not even thought of
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