I will suggest how probability ideas can help in making decisions in the face of uncertainty, and also describe circumstances where misunderstandings can arise.

Odds?

Recall that probabilities can be expressed in terms of odds, and vice versa: a probability of 1/5 is the same as odds of 4 to 1 against. Unfortunately, the term ‘odds’ has also been usurped by the gambling community to mean something quite different – the amount the bookies will pay if your selected horse wins. So when you read that Sea The Stars won the 2009 Derby at odds of 11 to 4, that simply means that for each ￡4 staked on the horse, the profit, because it won, is ￡11. The figures ‘11 to 4’ have no automatic relationship with the probability of winning. They depend on the bookies’ subjective assessments of the horse’s chances, and on how much money gamblers have staked. The term ‘payout price’ is a more accurate use of language for these figures like 11 to 4, but, regrettably, we have to accept the common usage of ‘odds’ in this gambling context.

A payout price is termed　fair if it gives no monetary advantage to either party, i.e. the mean value of the gamble is zero. The fair payout price for correctly picking the suit of a card selected from a well-shuffled deck is 3 to 1, as those are the exact odds against a correct guess.

Commercial gambles are not fair, in this sense, as they could never operate without a house advantage. For roulette in a UK casino, when all 37 outcomes are equally likely, the payout price for betting on a single number is only 35 to 1, not the 36 to 1 that would be fair. So the　mean return on a bet of ￡37 is ￡36, giving a house advantage – the percentage of any bet it expects to win – of 1/37, about 2.7%.

This advantage is the same for most of the available bets in roulette: whether you are betting on pairs of numbers, triples, groups of four, or six, or twelve, for every ￡37 you stake, your mean return is always ￡36. But in Las Vegas, the standard house advantage is bigger, because of an additional slot, double zero, giving 38 outcomes – with the same payout prices as in the UK. The mean return on $38 is generally $36, a house advantage of 2/38, or 5.3%.

A different way to bet on horse races is through a Tote or pari-mutuel system. Here, the money bet on all the horses is pooled, and a fixed proportion – 80% or so is common – is shared among those who backed the winner, in a size proportional to their stakes. The Tote advantage is then 20%, whichever horse wins.

The size of the bookies’ advantage in a horse race depends on which horse happens to win. Although bookies may make a loss or profit on any single race, recent data tell a sobering story: at a payout price of 6 to 4 on, punters should expect to lose about 10% of their stake; at a price of 5 to 1, expect to lose about 13%; at 10 to 1 the mean loss figure is over 23%, and if you speculate on horses priced at 50 to 1, expect to lose about two-thirds of your money.

This phenomenon is known as the　favourite-longshot bias . Punters lose their money　more slowly from bets on the more favoured horses than if they are attracted by large payout prices. Bookmakers were delighted when Mon Mome won the Grand National in 2009 at a price of 100 to 1.

Absolute risk, or relative risk?

Suppose that, among a particular group of people, the chance of developing colorectal cancer over the next five years is quoted as one in a thousand. We expect about ten among 10,000 people to develop the cancer. A new drug would reduce the chance to one in two thousand: then only about five among 10,000 would succumb if the new drug is used. The drug company could headline its press release ‘Risk of cancer cut by 50%’. And that is accurate: for each person, the risk would be halved.

This approach describes a　reduction in relative risk , and is often criticized as putting too favourable an interpretation on the data. For, suppose the initial risk had been one in ten million: cutting it by 50% leads to a new risk of one in twenty million, but in either instance, the risk is so small that among 10,000 people, we would expect pretty much the same number of cases – zero. Despite the risk being halved, the drug would hardly ever make a difference.

But suppose members of this group had a much higher chance, say 40%, of developing the cancer. A drug reducing the chance to 20% would qualify for the same headline, and would correctly be hailed as a major breakthrough, as among 10,000 people, fully 2,000 fewer would develop the cancer.

Rather than focus on the relative risk, it is usually more meaningful to look at the change in absolute risk. In the first case above, the absolute risk changes from 0.1% to 0.05%, so the drop is 0.05%; in the second case, the drop is a minuscule 0.000005%, while with the final figures, the drop is an impressive 20%.

A sensible way to proceed is to state the mean number of patients who should take the drug in order to prevent one case of the disease – the Number Needed to Treat , or NNT. The fewer the better, and this number is just the reciprocal of the change in absolute risk. In the examples above, the respective NNTs are two thousand, twenty million, and five.

Treating twenty million people to prevent one case of a disease is hard to justify. The NNT, along with knowledge of the treatment costs and the severity of the impact of the disease, allows us to make sensible decisions about allocating health care resources.

Combining tiny probabilities

How likely is it that at least one among an enormous number of events, each having a tiny probability, will occur? This can be the pertinent question when considering the probability of a catastrophe. Complex systems or machinery may fail if any one of a myriad of components fails; will two aircraft collide, or might a nuclear power station suffer a meltdown? The so-called Borel-Cantelli Lemmas give some pointers. These mathematical results show that, in many circumstances, the key quantity is the　sum of all those tiny probabilities: if it grows without bound, catastrophe is certain.

One consequence is that we can never be satisfied with current safety standards. It is essential to continue to make improvements.

For, no matter how high our standards, there is some non-zero probability of failure during any given month: and however small this value, if it remains unchanged (or even if it decreases too slowly) the sum over many months will grow indefinitely large, and disaster will occur sometime.

A sound programme of continual improvement does not guarantee that disaster will be averted: but to be ever satisfied with the status quo is to invite doom.

Some misunderstandings

(a) When a doctor tells a patient that there is a 30% chance that a particular medication will have unpleasant side effects, he means that he expects about 30% of patients on this drug to suffer. However, the patient may believe that these effects will arise on about 30% of the occasions on which she takes the drug. The doctor is thinking about all the patients he sees, the patient about all the times she takes pills – their reference classes are different.

(b) How does the public interpret the claim ‘There is a 30% chance of rain in Chicago tomorrow’ from a TV weather forecaster? The forecasters expect their audience to make a frequency interpretation, i.e. that, in the long run, rain would fall next day on 30% of the occasions when the weather conditions were similar to those now seen.

But when questioned, even among those viewers who were happy with the phrase ‘a 30% chance’, there was a spread of beliefs. Some felt that they were being told that rain would fall over 30% of the city’s area; others that it would rain in Chicago for 30% of the day; and some believed that 30% of meteorologists expected it to rain! A few thought that it would definitely rain, with the 30% figure indicating the rain’s intensity. There were many mismatches between the event the forecasters were referring to, and the event viewers were thinking about.

(c) If as few as twenty-three people chosen at random gather together, it is more likely than not that two of them share a birthday. When people meet this fact for the first time, they are normally surprised, but usually become convinced when proper counting shows this claim to be true. However, a minority remain unconvinced, because they mistakenly think they have been told that, if they and twenty-two others gather together, it is more likely than not that one of the others shares　their birthday. Listen carefully!

(d) Suppose that a coin, accepted as fair, shows Tails on nine consecutive tosses. Some will claim that the next toss is almost certain to be Heads, perhaps by invoking some ‘Law of Averages’ that requires Heads to eat into this excess of Tails immediately. No such Law exists. The Law of Large Numbers does imply equal proportions of Heads and Tails, but only in the long run : any sequence of nine Tails is diluted by the thousands of tosses before and after.

Alternatively, some will (correctly) note that the chance of ten consecutive Tails is less than one in a thousand; so if they see nine consecutive Tails, they may then ‘deduce’ that Tails next time is highly unlikely. But that is false logic: if ever we have nine Tails in a row, a tenth Tail will happen half the time. This confusion of the absolute probability of an event, with its conditional probability, given a certain background, famously arose in 1996: jockey Frankie Dettori rode the winners of the first six races at Ascot, and since no-one had ever won all seven races in a day, it ‘must’ be virtually impossible for Dettori to win the last race. But he did. Very few people will ride the first six winners in a seven race meeting, but when someone does so, he might well also win the last.

Ask yourself: am I assessing the absolute chance of twenty things happening, or just the conditional chance of the twentieth, given that the first nineteen have happened?

(e) Newspapers are often produced under great time pressure, so it is not surprising that some articles contain nonsense. But here are three reports that should have been spiked.

When all six eggs in a box bought in a supermarket were double-yolked, it was claimed that a truly astonishing event had happened. Only one egg in a thousand has this property, so the chance of getting a boxful is this tiny fraction, multiplied together six times. The resulting number is so small that if you opened one box every second, you would expect to take over thirty billion years to come across a box containing only double-yolked eggs!

But that calculation makes sense only if all the eggs in a box are chosen independently from a vast collection of eggs, one in a thousand of which is double-yolked. This doesn’t happen. Eggs are sorted by size before being boxed. Some boxes are even labelled as containing only double-yolked eggs . . . .

Allegations about the private life of an England soccer captain surfaced. A reporter offered figures for the ‘likelihood’ of each of four possible actions by the team manager:

(a) expel him from the playing squad – 1/10

(b) retain in the squad, but invite him to resign – 3/10

(c) retain in the squad, but remove the captaincy – 6/10

(d) take no action – 8/10

Any single one of these four estimated probabilities is plausible. But since they relate to mutually exclusive outcomes, their sum must not exceed unity: however, these ‘probabilities’ add up to 1.8.

Thirdly, it was reported that, among people who had won at least ￡50,000 on the National Lottery, the names turning up most often included John, David, Michael, Margaret, Susan, and Patricia. So far so good: but it was absurd to claim that　therefore you should try to include people with those names in your Lottery syndicate!

Describing ignorance

Drawing one card from a well-shuffled pack, I expect Black and Red to be equally likely, and I confidently attach the figure of ‘1/2’ to the chance of Red. In a slightly convoluted way, I could say that the　distribution I attach to the chance of getting Red allocates 100% to the quantity ‘1/2’ – my confidence is shown by the choice of 100%. If I am sure that a probability is some definite figure, I attach probability 100% to that figure.

But frequently, I cannot select a single figure: my best estimate of the chance my train will miss its connection may be 3/4, but 2/3 and 4/5 could be nearly as plausible, and even values close to the extremes of zero and unity are not completely ruled out. I can use a continuous distribution over the range from zero to unity to describe my feelings about this unknown probability.

Total ignorance of a probability – very rare circumstances – would be described by using the continuous uniform distribution of Figure 5. More often, there is some intermediate figure which is your best single guess at the probability, and your instinct about both higher and lower values is that their chances fall away towards zero.　Figure　8 shows a collection of graphs from what is called the　beta family of distributions that have this property.

Figure　8a indicates that we are pretty ignorant of the value of the probability: we attach highest belief to values near 1/2, but values as small as 1/5 or as high as 4/5 are still quite possible; with Figure　8b , we are much more confident that the value is close to 1/2, while still not ruling out the extremes. With 8c, our highest belief is for values close to 1/3, while with 8d the values are very strongly congregated near 2/3, with very little expectation that the value is below 1/2.

How far will my car travel on 10 litres of petrol? When warmed up, and driven at a steady speed, I expect to get about 90 miles, but if I make short journeys over a couple of weeks, it will be more like 60 miles. In either case, the distance will have some uncertainty, expressed via some continuous distribution. To get a handle on what kind of distribution, imagine splitting the petrol into small cups of size 10cc. There will be 1,000 such cups, and the total distance covered will be the sum of the distances achieved using these 1,000 components. Recall the Central Limit Theorem, which says that the　sum of a large number of random quantities will tend to follow a Gaussian distribution.

8. Beta family of distributions

8. Continued

For steady motorway travel, I would select a Gaussian distribution centred on 90, sharply peaked to show low variance. For sporadic travel within town, I would also use a Gaussian distribution, but centred on 60 and with a wider spread to demonstrate the greater uncertainty.

Utility

Your fairy godmother makes you a once only offer. Either she will give you ￡1, or she will toss a coin and if you call Heads or Tails correctly, she will give you ￡10, otherwise you get nothing. What choice would you make?

The alternatives are ￡1 for certain, or a fifty-fifty gamble with a mean payout of ￡5. Nearly everybody prefers the latter. But scale the money up by a factor of one million: overwhelmingly, preferences change. To have one million pounds for certain is far more attractive than being equally likely to have zero, or ten million pounds. The concept of　utility lies behind this difference.

For small sums of money, having twice as much usually　is worth twice as much, but if one million pounds would generate a certain level of pleasure for you and your family, double that amount would not lead to double the pleasure. In whatever units you choose to measure the ‘worth’ of a monetary amount, the usual shape of the relationship will follow that shown in　Figure　9 : the graph always rises, initially like a straight line, but then steadily more slowly.

‘Utility’ explains why a householder and an insurance company can agree that a sum of ￡250 is a reasonable annual premium to insure a house worth ￡250,000 against perils such as fire, subsidence, orflooding. The substantial reserves of the insurance company mean that, on any individual house, it can act as though the utility is identical to the relatively small sums involved. So long as the chance that it will pay out in any year is less than one in a thousand, its expected value for this transaction is positive, it will make a profit. On the other hand, the uninsured householder would face the enormous negative utility of finding ￡250,000 if she lost her house to one of those perils, so voluntarily surrendering just ￡250 to remove that possibility is a good deal to her too.

9. The general shape of a utility curve

Taking out insurance against breakdown of televisions, microwave ovens, and so on is almost always a bad idea. The sums are much smaller, utility and money are essentially the same, and the company will charge a premium large enough for them to expect a profit. Instead of buying this expensive insurance, build up a repair fund by placing the hypothetical premium into your bank account. Very few people will regret that move.

If you can successfully construct your own utility function, you can use it to help make a choice between the available actions under conditions of uncertainty. For each action, calculate the　expected utility of the outcomes, i.e. weight the utilities by their respective chances. Then select that action for which this expected utility is as large as possible.

That is the probabilist’s universal recipe for making the best of the available choices, whatever the circumstances.