the 1930s by a brilliant Italian mathematician, Bruno de Finetti, and fully formalised in the 1950s by American Jimmie Savage. Like frequentist probabilities, Bayesian probabilities require events with a known range of outcomes that cannot be influenced by the individuals estimating the probabilities. However, Bayesians understand that different people can have different information and opinions about events, and that some events cannot be repeated.
Bayesians define probability as subjective belief, measured by how much you would bet on various outcomes. For example, if you’re willing to bet $40 against $60 that it will rain tomorrow; and equally willing to bet $60 against $40 that it won’t rain tomorrow; your subjective probability that it will rain tomorrow is 40 per cent. Of course, other people can have their own views, which may differ considerably from yours.
The great virtue of Bayesian methods is that they can give direct answers to questions such as, ‘What’s the probability that it will rain tomorrow?’ According to strict Bayesian theory, different individuals can have different probabilities for the same event, but one individual never has conflicting beliefs. That is, the Bayesian who thinks the probability that it will rain tomorrow is 40 per cent cannot also believe that the probability that more than a centimetre of rain will fall tomorrow is 50 per cent. The second statement must have the same or lower probability than the first.
Like frequentists, Bayesians turn all risk questions into gambling games. But frequentists use games like dice and coin flips, in which everyone can agree on the probabilities, and the games are fair; not necessarily fair in the sense of having equal odds of winning, but in the sense that no one can influence or predict the outcome.
Bayesians, by contrast, use what a gambler calls proposition bets, bets about the truth of some proposition, such as ‘It will rain tomorrow’ or ‘Germany will win the World Cup’ rather than bets on a mechanical device like dice or cards. Wagers on sporting outcomes are often proposition bets. De Finetti’s famous example is a bet that pays £1 if there was life on Mars a billion years ago. Assume that an expedition will settle this question tomorrow, and consider the price at which you would buy or sell the £1 claim. If you price the claim at £0.05 and are willing to buy it for that price or sell it to someone else at that price, then di Finetti says that the probability of life existing on Mars a billion years ago is 5 per cent … to you.
Notice that this bet isn’t fair in the frequentist sense. Both sides are expected to do their own research and have their own opinions about the outcome. In many such bets, both sides are expected to attempt to influence the outcome – the simplest example is two competitors betting on the outcome of a match they’re about to play.
In place of fairness, Bayesians prize consistency. It’s entirely possible for two equally good frequentist models based on the same data to give different answers to the probability of rain tomorrow. But for a Bayesian, any individual at any given time can give only one answer to that question. Moreover, frequentist methods can give inconsistent results – for example, a probability of rain tomorrow greater than the sum of the probability of rain before noon tomorrow plus the probability of rain after noon tomorrow. That cannot happen for a Bayesian.
Playing Roulette
The mathematical study of risk began with casino gambling games. Later Bayesians did a reappraisal of that work using proposition bets in which you take one side or the other on a proposal (see the preceding section). Although you can get a great deal of insight from both approaches, they’re inadequate for risk management, even if you’re managing simple betting risk – the risk created for the purpose of making the bet.
Spinning the roulette wheel
You don’t need to understand the details of roulette for this discussion but I offer them here just in case you’re not familiar with the game. It may make some of the discussion clearer.
A roulette wheel consists of a wooden bowl with a spindle in the middle. The wheel head is a metal disk placed on the spindle, with the numbers 0 to 36 (plus 00 in American wheels) in slots around the outside. The slots are lined with a springy material – wood or plastic. The bowl has raised areas called deflectors that look like decorations but serve an important purpose.
The wheel head is spun in one direction, and a small, hard plastic ball is spun in the opposite direction against a lip at the outside of the bowl. As the ball slows, it drifts down from the lip, spiralling down toward the wheel head. Along the way, its direction gets changed when it hits deflectors. Eventually the ball touches the wheel head, which is still moving quickly enough to give the ball a hard bounce; also the springy wood or plastic in the number slots keeps the ball bouncing around for a while. But eventually the ball settles into a slot, and the wheel head slows to a stop.
Players make bets on the number the ball will land in, or its colour (18 of the numbers are red, 18 are black and the zeros are green) or other combinations. Most payouts are computed as the fair amount – the payout amount that would cause bettors to break even in the long run – ignoring the zeros. So if a player bets £1 on a single number, she wins £35 if she’s right and loses her £1 if she’s wrong. If you ignore the zeros, she wins one time for every 35 losses and breaks even. However, because she loses on zeros, she really has 36 (37 for an American wheel) £1 losses for each £35 gain.
Casinos allow bets for a brief period after the wheel is spun, which is what creates one of the main edges for advantage players (gamblers who play in way such that they, not the house, get the advantage). One reason is that a longer time for the bettors means more bets, which means more profit for the casino. But the other reason is that it makes the wheel seem more fair, because the croupier (the casino employee who spins the wheel) cannot try to bias the spin against your bet. Many players prefer to bet only on things that are in spin, meaning no human can influence the outcome any more.
Winners are paid, losing bets are collected and the next round begins.
Blaise Pascal, the 17th-century French mathematician who was one of the two founders of mathematical probability theory, actually invented the first roulette wheel in 1655 (about a decade before he got interested in probability) while trying to build a perpetual motion machine. But roulette didn’t get popular as a gambling game until nearly 150 years later. Almost immediately afterwards, however, gamblers got the idea that they could beat the game by exploiting biased wheels. That meant they observed results looking for numbers that came up more often than average, due to tilting or another defect in the wheel.
It took another 150 years for the next big insight by Ed Thorp, who realised that if the wheel was biased, of course you could beat it. But he appears to be the first person to also realise that if the wheel were not biased, it had to be machined so well that it would be predictable. This insight is a fundamental one about risk in general, not just roulette.
This point is obscured by the English language. When the average person says a roulette wheel is random, she means each number comes up with equal probability – there’s no advantage to betting one number over another. When a statistician says the roulette wheel is random, she means the numbers are unpredictable – not the same thing at all. If the roulette ball landed in the numbers 1, 2, 3 and so on in order up to 36 and then back to 0 (or 00 on American wheels), the outcome would be perfectly random in the first sense in that each number would come up the same number of times. But the outcomes would be completely non-random in the second sense in that the outcome was highly predictable. If the roulette wheel always came up 1 or 2, but mixed the two numbers perfectly, the wheel would be non-random to the average person, but perfectly random to the statistician.
You can easily build a roulette wheel that’s unpredictable – any kind of sloppy engineering will do. But you’re likely to find that this wheel is also biased and that some numbers come up more than others. If you machine the wheel so perfectly that the numbers all come up with exactly the same frequency, it’s likely that someone observing the spin can predict where the ball ends up – not necessarily