The vital importance of understanding the improbable
27/04/2010 - 11:43
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Imagine you are taking part in a TV game show. The games master shows you three, closed doors. Behind one of them is the prize of a brand new car; behind each of the other two, a goat. The games master asks you to choose one of the doors, but before he allows you to open it, he, knowing where the car is, opens one of the un-chosen doors revealing a goat. He then gives you the opportunity to change your mind. You can either stay with your original choice or switch to choose the other closed door.

 

At this point you would probably feel that it would be a 50:50 choice.  That you had equal chances of winning which ever option you took. You would be wrong. To improve your chances of winning, you should switch. (See box for explanation)

 

The Monty Hall problem

This problem, known as the "Monty Hall" problem, was first published in a puzzles column of the American magazine "Parade" in 1990. The answer, that you should switch, stimulated a storm of protest. Around 10,000 letters were received denouncing the journalist, Marilyn von Savant, for giving a stupid and wrong answer.

Everyone believed the chances of winning were 50:50. Writers of these letters included one thousand PhDs, some of whom were mathematicians.  One eminent mathematician only believed the correct solution when he was shown a computer simulation of the game.

The fascinating thing is that this problem is very simple, and yet our intuitive answer is wrong. It illustrates how poorly our brains are wired to tackle probability. This disability seems to be a consequence of evolutionary past (see XAD - Evolution of the Mind). It has far reaching impacts on how we perceive the world and in our daily lives.

 

Randomness rules our lives

In his fascinating book "The Drunkard's Walk - How Randomness rules our lives," physicist Leonard Mlodinov analyses situations where we are misled. Our intuitive brain is wired to recognise patterns (even where they do not exist), and is fooled by chance and randomness. In this article we shall take a look at two facets of life where a wrong intuitive assessment of chance can have far reaching consequences: medical screening and court cases.

 

The paradox of the false positive

Imagine you are invited to take a blood test to screen for a fatal disease. This test is 99% accurate (that is, one percent of the test results are wrong, showing positive when the person does not have the disease (false positive) or negative when they are ill (false negative).One in 10,000 people are known to have the disease in Spain. You take the test and the result comes back positive.

 

What will your doctor say to you? a)...sit down, I have some bad news for you or b) ...don't worry, the risk is small. The answer is b). Here is a rough explanation:

"99% accurate" means that if 10,000 tests are done, 1% of them, that is 100 test results, will be wrong. The wrong results will probably be false positives because the disease is quite rare - only one in 10,000 people is actually infected, and there will only one truly positive test for every 10,000 people tested. So, in every 10,000 people tested there is likely to be around 99 false positives and one actual positive.

Thus, even though you have a positive test, you have a 99% chance of NOT having the disease. This situation is known as the "Paradox of the false positive," and it occurs when the sensitivity of the test is not good enough in relation to the rarity of the condition.

 

Even physicians are fooled by this paradox. In studies in Germany and the US, doctors were asked to estimate the likelihood that an asymptomatic woman between the ages of 40 and 50 had cancer if her mammogram was positive. They were told that in 7% of mammograms, cancer was seen when there was none and that actual cancer incidence was about 0.8%; and false negatives (missed cancers) were about 10%.

One third of the German doctors reckoned that it was 90% probable that the woman had breast cancer, while 95 out of 100 American doctors thought the probability was 75%. In fact the probability is around 9% (It is worth writing down both the above examples and calculating them to prove this to yourself).  

 

This tendency to underestimate false positives can result in devastating changes to people's lives.  For each HIV positive person, each athletics drugs cheat, and each terrorist detected there may be ten, hundreds or thousands of people who have been wrongly identified. Currently, there is international controversy over the value of breast cancer screening by mammogram. The high proportion of false positives means than many women undergo unnecessary treatment, and some research indicates that breast cancer screening programs may actually have no effect on mortality.

 

The O.J. Simpson trial

Lawyers are especially clever at exploiting our misunderstanding of probability. In the famous O.J. Simpson trial, where an American football star was accused of murdering his wife, it was acknowledged that Simpson had a history of domestic violence. However, the defence presented the 1992 US statistic that while four million women were battered annually by their partners, only 1,432 (or one in 2,500) were actually killed by their abusers. Thus they implied that even though O.J. may have slapped his wife, the chances that he murdered her were very small.

Unfortunately, the prosecution (which was outmanoeuvred throughout the trial) failed to pick up on this flawed logic. The important number is not the proportion of battered wives who are murdered by their partners, but the proportion of murdered battered wives whose murderers were their partners (as opposed to a third person). The answer is 90%.  This statistic was not mentioned at the trial. Simpson was acquitted.

 

Mother convicted of killing her sons

More recently, the incorrect use of probability and statistics in a UK case led to the wrongful conviction of a mother for the murder of her two baby sons. In 1996, Sally Clark's 11-week-old son Christopher was found dead in his cot. The death was ascribed to "Sudden Infant Death Syndrome" (SIDS), a label given when a child dies and there is no known cause.

She conceived again and early in 1998, eight-week-old Harry collapsed and died. Sally was arrested and accused of murdering both her children by smothering them. At trial, an expert paediatrician and child abuse specialist, Sir Roy Meadows, testified that the odds of both children dying from SIDS were 73 million to one. He arrived at this figure by taking the odds that a child would die of SIDS, which are 1 to 8,543 and multiplying them together. The jury, adopting the view that "lightning never strikes twice", convicted the mother.

Sir Roy's calculation was flawed. He assumed that the deaths were independent and that there were no genetic or environmental factors which might increase the risk of a second child dying of SIDS. The other logical flaw in the case is similar to that in the O.J. Simpson trial. What we need to know is not the probability that two children will die of SIDS, but the probability that the two children who died, died of SIDS or, conversely, died of murder.

It's the relative likelihood of these two scenarios which is important. A mathematician has since calculated that two infants are nine times more likely to die of SIDS than to be murder victims. Sally remained in prison for three years, and her conviction was only quashed after two appeals. She was finally released in 2003 to join her husband and young son, and to attempt to resume her career as a solicitor. In 2007 she was found dead of acute alcohol intoxication, aged 42.

 

The case of the killer nurse

Another such miscarriage of justice has only just been resolved. In 2004, Dutch nurse Lucia de Berk was jailed for life for the murder of seven patients and the attempted murder of three more. None of the alleged victims had undergone post mortem examinations. The statistical probability of her being present at so many deaths was central to the prosecution's case.

As in the Sally Clark case, the probabilities of these people dying by chance when Lucia was on duty were multiplied together. The idea that this was a coincidence and run of bad luck was shown to be unlikely, even though most of the patients were very old and very sick. Again, as in the Sally Clark case, the statistics and probabilities were wrongly applied. The court should have weighed up the probabilities of the two different explanations: murder or coincidence. 

The argument that the deaths were unlikely to have occurred by chance was not important. What really mattered was the relative likelihood of the two explanations. However, the court was given an estimate for only the first scenario. The case was finally overturned in April 2010 following a re-trial.

Lucia is penniless, having been denied unemployment benefits because of her unusual status, and paralysed down one side following a stroke which she had in 2006, aged 44, in the week she was told that her conviction would be upheld. The only good thing is that she now has the opportunity to claim substantial damages.

 

Thus, another life has been ruined. Some would say, as the result of stupidity. However the real culprit is our inability as human beings to visualise easily the consequences of probability and statistics, combined with our inherent tendency to see patterns where there are none. We know that "shit happens" -- but sometimes we find it hard to believe.

 

 

by Christine Betterton Jones - BSc. (Zoology), PhD (Parasitology)

 

 

Bibliography

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