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Whether you clean out the bank while gambling at Monte Carlo, contract leukaemia, or are run over on a zebra crossing by a drunken driver, risk is an integral part of our daily lives

THOUSANDS of people died in the Armenian earthquake of 1988. We shall probably never know the precise total. Nearly 200 people died in the San Francisco 鈥榪uake last year. When the Herald of Free Enterprise turned over in the cold waters outside Zeebrugge in 1987, the death toll eventually reached 193. Less headline catching is the routine death of 15 people, on average, every day on Britain鈥檚 roads. All human actions 鈥� including inaction 鈥� carry an element of risk.

How much risk depends on the activity. Leaping across a canyon on a motorcycle carries a large degree of risk; watching television in the living room a small one.

Politicians may say that the risk of a particular accident is one in a million. To understand this statement you need a clear grasp of the scope and the limitations of the mathematics that underlie it. You also need to know what assumptions it is based upon.

Taking a chance

The meaning of risk

THE DEGREE of risk is the extent to which harm may result from a particular action. There are many different kinds of risk, including economic risks, the risk to life and health and the risk to the environment. Risk is not the same as danger. Walking downstairs carries a definite degree of risk: it is one of the commonest causes of accidents in the home. But you would hardly call it dangerous. In our daily lives we all risk being involved in accidents. If you ride in a car, it may crash. If you choose to walk, a car may hit you. If you stay at home and light the gas, the house may catch fire.

Medical science has reduced the risk of catching a serious disease, but it has not eliminated it 鈥� people still suffer from AIDS, legionnaire鈥檚 disease and food poisoning. Because it is impossible to eliminate risk, the best that we can do is to balance risk against benefit. Far more people would die of cold if the government banned gas fires than die in house fires or gas explosions because of them. So the benefit greatly outweighs the risk, and the decision is easy.

Nuclear power is less straight-forward. There are benefits. Nuclear power produces less acid rain than does the process of making electricity from burning fossil fuels. Producing nuclear electricity does not involve the death of a single coal miner. There are risks, however, such as pollution or a catastrophic release of radiation in a major accident, as well as deaths among uranium miners. The mathematical analysis of risk provides objective and rational methods for making such judgements. But just because a method is objective and rational, that does not mean it is necessarily right. The mathematics depends upon assumptions about the real world, about how people behave, and about how accurate information is. The mathematics of risk is an aid to judgement, not a substitute for it.

Probability is a way of expressing risk mathematically. It is always a number between 0 and 1. An impossible event has a probability of 0; an event that is certain to happen has probability of 1. Everything else lies somewhere in between. Probability is the proportion of cases in which an event occurs. For example, if you toss a fair coin, you expect to get heads about as often as tails. In the long run, about half the tosses will be 鈥渉eads鈥�. We say that the probability of the event 鈥渉eads鈥� is a half. The probability of rolling a six with a fair die is one in six. We can write these either as fractions (1/2, 1/6) or as decimals (0.5, 0.167).

According to the old Central Electricity Generating Board, the probability of a catastrophic accident in a nuclear power station was one every 10 000 years. Catastrophic means the release of large amounts of radioactive material into the environment. A probability of one every 10 000 years sounds very reassuring, but it is worth taking a closer look. What it means is that for each nuclear reactor, the probability of a catastrophic accident in any given year is one in 10 000; that is, 0.0001 per year. There are roughly 40 nuclear power stations in Britain, so the probability that at least one will have a catastrophic accident in any given year is the sum of the 40 probabilities, which is 0.004. The probability of at least one catastrophic accident in Britain during the next 25 years is 25 times this, or 0.1. That is, the chances are one in 10. This does not sound as reassuring as 鈥渙ne every 10 000 years鈥�. But it is just a different way of saying the same thing.

Great expectations

Pools losers

WHETHER or not a risk is acceptable depends not only on the probability of a harmful event, but also on how harmful it is. Few people would worry if they were told that with probability 0.5 they might lose 10p. They would be less happy if the sum involved was a 拢1 million. The simplest way to deal with the scale of loss is to multiply the probability of the event by the amount that could be lost. This is called the expected loss. In these two examples the expected losses are 5p and 拢500 000.

Cost-benefit analysis, or risk-benefit analysis, is a way of balancing the expected loss against the risk. In everyday life, people often take decisions where the expected benefit is negative. On average, people who take out life insurance policies or play the football pools will lose. To insure a life, people pay a certain amount of money, the premium, to the insurance company. In return, the company guarantees to pay a much larger amount when the insured person dies. The insurance companies have to make a profit to stay in business, so they employ actuaries (statisticians who specialise in calculating risks) to set the premium high enough to guarantee that on average the companies wins. This means that each person they insure has an expected loss. Despite this, it makes sense to take out life insurance. The cost of the premium is relatively small, something that the person involved can easily afford to pay. The benefit if that person dies unexpectedly is large: the person鈥檚 family will still have a house to live in and enough money to buy food and clothing.

A similar analysis applies to football pools, and bookmakers. On average, only the pools company gains. But individuals can risk tiny amounts and have a small chance that a big win will transform their lives. The main difference between the pools and the insurance is that in life insurance the aim is not to win the jackpot.

One problem with 鈥渆xpected value鈥� is that it treats everything alike in all circumstances. It considers a loss of 拢1 to a pauper to be the same as a loss of 拢1 to a millionaire.

We can measure things differently. If, instead of the average gain or loss we consider the maximum gain or loss, then the picture for life insurance changes dramatically. For a premium of a 拢10 per month over a period of 5 years, you can buy insurance cover of 拢20 000. The maximum loss is 拢600 (拢10脳60 months); the maximum gain is 拢20 000, less any premiums you might have paid.

Of course, it is no more clear that the maximum, rather than the average, is the 鈥渟ensible鈥� measure. Mathematics can provide a limitless range of such measures. But mathematics alone cannot tell us which, if any, is the 鈥渞ight鈥� one to use. There may be no right measure. In fact, one of the difficulties in the debate over nuclear power may be just this point. No matter how small the risk of a really serious accident, the effects if one were to occur would be horrifying. Is it possible to do a calculation that will balance any gain against the tiny chance of killing a million people?

鈥楺uake hits Manchester

Calculating the odds

EVERY DAY, commercial aircraft make an enormous number of flights; every year a few of them crash. We can estimate the probability of a crash by dividing the number of crashes by the total number of flights. The more often an event occurs, the more accurately we can estimate its probability.

It is more difficult to estimate the probability of a rare event in this way. For example, what is the probability of a severe earthquake happening in the Manchester area?

No one has ever recorded such an event, so we might estimate the probability as 0. But this must be an underestimate. Even though earthquakes are rarer in Britain than they are in other parts of the world, such as California or Japan, they do happen. In April 1990 the second largest earthquake in Britain this century, registering 5.2 on the Richter scale, hit Clun, in Shropshire. The probability of a big earthquake in Manchester is very low, but it is extremely difficult to say how low.

Unexpected sources of risk cause even more problems. Before manufacturers began to use chlorofluorocarbons (CFCs) in aerosols, they investigated the likely effect of these chemicals on the environment, including possible damage to the ozone layer. The researchers chose CFCs because they are unusually stable compounds, and so are unlikely to react with atmospheric ozone. Unfortunately, nobody realised that ice crystals in the upper atmosphere would make their reaction with ozone much more likely. If your analysis of risk omits a major hazard because you do not have the imagination to consider it, or the information to work out its effects, then your analysis may be misleading.

The way in which researchers collect data and analyse them is also important. Averages can conceal more than they reveal. Britain has occasional high winds. When the wind gusts above 60 knots it can damage buildings. The average wind speed is less than 10 knots. However, this does not mean that the wind will never cause structural damage. In three out of the past 30 years, wind speeds at Heathrow airport have exceeded 60 knots, including the great storm of 1987 which damaged buildings and uprooted trees over a swathe of southern Britain.

Another example is assessing the rate at which AIDS is transmitted. Several traditional methods work with averaged figures for infection, rate of sexual contact, and so on. But the risks differ widely for different groups, because they depend on lifestyles. Using averages in these circumstances may produce nonsensical figures.

Another problem is finding out whether a hazard really is the cause of some adverse event. Ten years ago, people argued fiercely about whether cigarettes cause lung cancer (some tobacco companies still deny that this is true). As recently as six years ago the British government was disputing that lead in petrol is dangerous, even though lead is known to impair children鈥檚 intelligence.

A topical example of this problem is leukaemia clusters. Leukaemia is a fatal cancerous disease of the blood that affects children in particular. In some parts of Britain there is an unusually large number of leukaemia cases. Researchers have discovered some of these 鈥渃lusters鈥� near nuclear power stations, and related nuclear installations, such as Sellafield. Does this mean that nuclear power stations cause leukaemia? Until recently that was unclear. First, some clusters are well away from nuclear power stations. Secondly, some nuclear power stations do not have clusters nearby. And as the number of leukaemia cases involved is relatively small, chance effects may also be operating. This makes it extremely hard for researchers to decide what is really going on.

In February 1990, however, Martin Gardner, of the University of Southampton, concluded that there was a link. He said that the exposure to radiation of men working at Sellafield appears to have caused genetic damage in children they fathered. British Nuclear Fuels at first advised workers exposed to the highest radiation not to have children, although it has since withdrawn this advice. In this case the link between radiation and children seems to be through the father, which explains why previous studies failed to find a direct connection between the clusters and the power stations.

This example shows how important it is to distinguish between correlation and causality. Two events are correlated if one is often accompanied by the other. To find a correlation is easy: you just see how often the two events occur together. Causality is one event causing another. To prove causality you must pin down the entire chain of events, and that can be very difficult.

Companies often complain that evidence that their product is unsafe 鈥渄oes not establish causality鈥�. Although correlations may not prove causality, they certainly do not disprove it.

In fact, all scientific evidence about causes is really about correlations. We observe in hundreds of experiments that one event leads to another. For example, litmus paper turns red when we put it in acid, and we assume that putting it in acid is what turns the litmus red. This is just a very high degree of correlation; it could all be coincidence. In the end we have to use some common sense.

Acceptable risk

A sporting chance

SOMEONE must decide at some stage whether or not a risk is 鈥渁cceptable鈥�. But what is an acceptable risk?

There is a bad answer: a risk is acceptable if we get the benefit while others suffer the effects. Dumping my toxic waste is fine provided you keep it well away from me. A drug is 鈥渟afe鈥� if I make the profits by selling it and other people run the risk of using it. All too often, decisions may be biased 鈥� perhaps unconsciously 鈥� by this kind of reasoning.

A better answer is that the benefits must outweigh the risks for most of the people involved. Most people think that the convenience of car travel outweighs the very distinct risk of being involved in a serious accident. People who indulge in 鈥渄angerous鈥� sports, such as mountaineering consider that the fun outweighs the risk 鈥� or perhaps they underestimate the risk.

The way people react to a risk does not always reflect its probability. For example, the probability of being killed in a terrorist attack on an aircraft is smaller than the probability of dying because the shuttle bus crashes on the way to the airport, but most travellers are more worried about the terrorist attack. The probability of getting rabies, even in countries where it is common, is very low 鈥� a few cases per million people per year. Yet the British government devotes much effort and money to very strict quarantine regulations to keep rabies out. Is this sensible, or a waste of money?

Mathematics does not answer this question. It can give us a good idea of the dangers involved in some activity, and mathematical assessments of levels of risk provide useful information for public debate. But, the mere fact that a mathematical calculation produces a very small figure for the level of risk does not mean that the risk is automatically 鈥渁cceptable鈥�.

For example, asbestos fibre can cause fatal lung disease. The probability of contracting such a disease is very low. But that does not mean we should go on using asbestos for ceiling tiles or brake linings.

In any case, very low may not be as low as it sounds: in a population of 50 million, a disease with an annual probability of one in a million will kill 50 people every year.

Again, we have to use our judgement and common sense: the numbers alone cannot take the decisions for us.

Before the Challenger disaster, in 1986, the officially estimated figure for an accident involving a space shuttle was one in 100 000. This calculation was ridiculously optimistic. It failed to take account of the way events might depend on one another. In particular, launching a shuttle on a cold day substantially increased the probability of several different events. In the subsequent inquiry, the American physicist Richard Feynman made the point that the calculations looked very fishy. An engineer eventually said the true figure was around one in 300.

Another problem with many methods of risk assessment is that inaccuracies in the figures can change the result of the calculation. This is important. In practice the estimate for a single event can easily be 10 times too large or 10 times too small, and the final answer may vary wildly.

One method for studying the effects of possible errors is called sensitivity analysis: it aims to determine the range of probabilities that a calculation of risk may yield.

Human psychology

Fear and dread

THE WAY that human beings react to risk is influenced by psychological factors. 鈥淔amiliarity breeds contempt鈥�, the old saying goes, and it might. Workers in hazardous professions often fail to take precautions. They are so used to the danger that they no longer fear it. In much the same way we fear the unknown. We often think unfamiliar hazards are more risky than they really are.

Even taking precautions may not always be as beneficial as we think. One theory, known as the risk compensation hypothesis, says that safety precautions may increase the exposure to risk. For example, a trapeze artist may take greater risks when performing with a safety net than without one. When drivers wear seat belts they may feel safer, hence drive less carefully: the end result could be more accidents rather than fewer.

People often assess probabilities wrongly. A common example is the law of averages, according to which lightning never strikes in the same place twice. In war, soldiers often shelter in shell craters on the grounds that a hit by a second shell is less likely. Neither of these beliefs is true.

Lightning is more likely to strike the same place again, because it tends to favour exposed places on high ground.

Artillery shells do not remember where previous shells have gone. (Though sheltering in a shell crater is perfectly sensible, because you can keep your head below ground level.)

Perhaps the best exposure of the 鈥渓aw of averages鈥� is the story of the man who always took a bomb with him when he travelled by plane, on the grounds that two bombs on the same plane is such an improbable occurrence that in practice it could never happen.

The great railway disaster

DISASTERS very rarely have a single cause. Risk involves chains of cause and effect, in which series of individual events combine to produce a disaster. In order to calculate the combined risk, it is important to have accurate estimates of the probabilities of the individual events. One technique that is used widely in risk assessment is to construct a fault tree. This is a diagram that shows the possible chains of events leading to a harmful outcome.

A simple example is parachuting. Inside each parachuter鈥檚 pack is a main parachute and a reserve. The jump is fatal if both parachutes fail to open. So the fault tree is a two-link chain. If the probability of failure is one in a thousand for each parachute then the total probability is one in a million.

For a slightly more complicated example, consider a train passing a signal. An accident may happen if either the signal fails or the driver ignores (or misreads) it. But that fault alone will not cause an accident: there has to be an obstacle on the line, such as another train. If the obstacle is close, then the train will hit it. If it is further away, the train may have time to stop 鈥� unless the brakes fail. With some rather arbitrary figures for probabilities, the fault tree looks like the diagram.

For the train we must trace all possible chains through the tree, work out their probabilities by multiplying, and adding the results together.

Two chains of causality contribute almost all of the probability here: signal failure/obstacle on the line; or, driver error/obstacle on the line. So for this particular fault tree, brake failure has little significant effect.

These calculations contain several hidden assumptions. The main one is that each branch of the tree involves mutually exclusive events which are independent of each other.

Probability rules

Rule 1: If probability of an event A is p(A), then the probability that A does not happen is:

p(not A) = 1-p(A).

Rule 2: Two events are mutually exclusive if they cannot both occur together. For example, if you throw a die, then the events 鈥渁 six鈥� and 鈥渁 five鈥� are mutually exclusive, because you can not throw both at once. The probability that one or the other occurs is the sum of the separate probabilities:

p(A or B) = p(A) + p(B).

Rule 3: Suppose two experiments occur in succession. They are independent if the outcome of the first has no effect on the outcome of the second, such as tossing a coin and then drawing a card from a pack. What happens to the coin does not affect the drawing of a card from the pack. You can find out the probability that both will occur by multiplying the separate probabilities:

p(A and B) = p(A) p(B).

Rule 4: This is an approximation that mathematicians often use in risk calculations. Suppose that A and B are events, independent but not necessarily mutually exclusive, whose probabilities p(A)=a and p(B)=b are very small. What is the probability that at least one of them happens? It should be the sum of the following probabilities:

p(A and not B) = a(1-b).

p(not A and B) = (1-a)b.

p(A and B) = ab.

Thus:

p(A or B) = a(1-b) + (1-a)b + ab = a + b 鈥� ab.

If a and b are small then ab is very small. So we can neglect the term -ab, and so:

p(A or B) = a + b = p(A) + p(B).

In other words, events of small probability can be considered as being mutually exclusive, even if strictly speaking they are not independent of each other.

Further reading

Risk and Chance, edited by Jack Dowie and Paul Lefrere (Open University Press, Milton Keynes, 1980). Acceptable Risk, by Baruch Fischhoff, Sarah Lichtenstein, Paul Slovic, Stephen L. Derby, and Ralph L. Keeney (Cambridge University Press, Cambridge, 1981). How to Tell the Liars from the Statisticians, by Robert Hooke (Dekker, New York, 1983). Risk Assessment: a Study Group Report (Royal Society, London, 1983). Lady Luck, by Warren Weaver (Dover, New York, 1963). What do You Care What Other People Think? by Richard P. Feynman (Unwin Paperbacks, London, 1988). The BMA Guide to Living with Risk (Penguin, London, 1990).