Gladiators, Pirates and Games of Trust

by Haim Shapira

  The problem with the average is that it’s very sensitive to extreme values. If our manager doubled only his own salary, the average salary would almost double too, though no one else had gained a red cent. However, the median (remember that the median is the ‘middle’ value in the list of numbers ordered from the lowest to the highest) creates the opposite problem. That same salary hike for the manager would have no impact on the median, because the median is completely insensitive to extreme values. Therefore, if we want to present a situation in a numerically reasonable way, we must present both the median and the average, the standard deviation and the shape of the distribution. Interestingly, when wage data are presented in the news, what’s reported is almost always the average salary, or the average expenses of the average family (you understand the reason by now, I hope). Clearly, news editors feel they shouldn’t go into further statistical complications. That would only make their viewers switch channels, but you – the viewers – mustn’t draw any conclusions based on these data. Clearly, a statistician who has one foot immersed in icy water and the other in boiling water feels wonderful (on average).

  AN AVERAGE TREASURER

  I once read a report about a finance minister of a certain country who was cited as saying that he hopes that one day all workers in his country will start earning more than the national average (sometimes this ‘wise’ saying is attributed to Bill Clinton). I have to admit it’s a brilliant idea. All we can do is wish that treasurer a very long life – he’ll need it if he’s going to wait for that to happen. Reacting to that report, one reader stated that the treasurer doesn’t understand what an average is, and kindly explained: ‘50 per cent of employees earn more and 50 per cent earn less than the average.’ Naturally, he too wasn’t a big expert in statistics: he confused average with median.

  AVERAGE DRIVERS

  In another case, I read an article by a journalist who was supposed to know a thing or two about statistics. Stating that everyone believes they are better-than-average drivers, the journalist explained that it’s mathematically impossible for the majority of drivers to be better than the average. He was wrong. Here’s a simple explanation why. Suppose four of five given drivers had one traffic accident each last year, while the fifth was involved in 16 accidents. Between them, the five drivers had a total of 20 accidents, and the average per driver is four. Thus, four of the five drivers (that is, 80 per cent) are better than the average! Next time, when you read that almost everyone believes they are better-than-average drivers, don’t dismiss the claim so fast. Who knows? Perhaps they’re right (statistically speaking, at least).

  SPEAK FOR YOURSELF

  One of the strangest and most interesting things about statistics is the fact that many people who never studied the subject believe they understand it (show me a person who never studied partial differential equations or functional analysis and still claims he knows all about it). People often say things such as, ‘The numbers speak for themselves.’ This is silly. I’ve never heard the number 7 speaking for itself or conversing with the number 3. Have you?

  FUN READ

  In this context, I’d like to mention two of my favourite books. The first is A Mathematician Reads the Newspaper, in which John Allen Paulos explains how he (a mathematician) reads news items differently from the average (median) person. The other is a wonderful book by Darrell Huff, named How to Lie with Statistics. I often use this book when I start teaching statistics: it helps the students hate the subject a little less.

  Chapter 11

  AGAINST ALL ODDS

  In this chapter we’ll find out what we are talking about when we talk about chances. We’ll flip coins and roll dice, discuss the meaning of probability on operating tables, help physicians not to come up with wrong diagnoses, and try to pass a lie detector test without our lying being detected.

  THE DARK SIDE OF THE COIN

  At first glance, it seems that the concept of ‘chances’ or ‘odds’ is rather simple, and people indeed often say things such as ‘There’s a good chance that it will snow tomorrow’, ‘The chances that I’ll start exercising in the next 45 years are quite slim’, ‘The odds of seeing a 6 in dice games are 1 in 6’, ‘The chances of war next summer have just doubled’ or ‘He’ll probably not recover from this.’ Yet when we begin to explore and study the concept, it turns out that it’s much more complicated and confusing.

  Let’s begin with the simplest example: the toss of a coin. Any person asked about the odds that a tossed coin will reveal the heads side would naturally say that about half the flips are likely to show heads. This is probably the correct answer, but confusion sets in as soon as we ask, ‘Why did you say “half”? What knowledge is this reply based on?’

  When I teach probabilities or lecture about the subject, my listeners always give the same answer: ‘There are only two options – heads or tails – so the odds are 50-50, or half the times for each toss.’ Here I make their lives a bit harder and offer another example. Since we’re speaking about probabilities, Elvis Presley may walk through the door and sing ‘Love Me Tender’ for us, or he may not. Again, there are two options, but I wouldn’t say that the odds are 50-50. We could think of less enchanting things. Right now, as I write these words, the ceiling above my head might collapse and crumble; but then again, it may not. If I believed that the odds are 50-50, I’d run out of the room right now, even though I enjoy writing. In another example, a friend of mine had his tonsils removed. Again we had two options: either he’d survive the operation or not. All of his friends were hopeful for a happy ending, being almost certain that the odds were better than half in this case too.

  We could come up with many such examples, but the principle is clear: the fact that there are two options doesn’t guarantee 50-50 chances. Even if this idea is genetically embedded in our minds, the close association between two options and 50-50 odds is almost always wrong.

  So why do people say that when we toss a coin, the chances are that it will fall on either heads or tails half the times? The truth is that there’s no way of knowing that for certain. This is not one of Franklin’s certainties. If we want to verify that the chances are ‘half ’, we should hire someone who has just taken early retirement and therefore has plenty of time to spare, give him a coin, and ask him to toss it many, many times (we could explain this as occupational therapy). We must try this plenty of times because if we were to toss the coin only eight times, for example, we might come up with all kinds of results. It could be 6 heads and 2 tails; or 7 heads and 1 tail, or vice versa; or 4 of each; or any other combination. Still, if I tried that 1,000 times, the resulting ratio would probably be close to 1:1, or about 500 times each. If, instead, the results were 600 heads and 400 tails, we might suspect that the coin was somehow damaged and the chances it would fall on one side more than the other were higher because of that flaw. In this case, we could assume that the odds of the result being heads would be roughly 0.6.

  As we just saw, even an object as simple as a coin could cause problems, and we haven’t even started asking the big questions. We could ask, for example, why is it that if we toss an ordinary coin one thousand times, the result would be very approximately 500 heads and 500 tails? After all, coins have no memory, and no one toss is affected by the previous toss. I mean, after that coin has presented heads four times in a row, it wouldn’t think: ‘OK, that’s enough. Time to diversify and balance things out.’ Why shouldn’t we have a very long line of heads? Why do numbers tend to even out? (Food for thought.)

  DICE GAMES

  A partial explanation of the convergence to a predictable pattern of coin toss results can be found in the following story. A guy (whose name I won’t mention here) was asked to roll a die (singular of dice) 100 times and report his results to us. He told us that he’d hit 6 every time. We didn’t believe him, of course, but if he’d told us that his score had been 1, 5, 3, 4, 2, 3, 5 and so on, we could have believed him. We would even wonder why he bothered telling us abou
t this random set of figures. That outcome is so boring! And yet the chances of scoring either of these two specific results are identical. In fact, the odds of obtaining each of these two outcomes are precisely one-sixth to the power of 100, which is almost zero. (This too could make us start wondering why things that have a near-zero chance of happening still happen. This is a very broad-spectrum question because, observed from a great distance, almost everything that happens to us – starting with the very fact that we were even born – shouldn’t have happened at all, and yet it did.)

  So why do we not believe in a sequence of 100 sixes and yet find the second sequence totally plausible? Which is more likely in the first roll of the dice, 6 or 1? Obviously, there’s no difference. What about the second roll? Which is more likely, 6 or 5? Again, there’s no difference, and the odds of both are the same.

  What’s going on here?

  There’s potential confusion here, because we seem to be talking about a pure sequence of sixes, which indeed is very hard to score, and a mixed sequence, which is very easy. Yet a specific mixed sequence is as rare as a pure sequence.

  PROBABILITIES ON THE OPERATING TABLE

  Let’s consider a medical example. Suppose the chances of success for a certain surgical procedure are 0.95. What does that mean? First, we should understand that when discussing the success rates of operations (and similar issues), we should have as large a sample as possible. Such a probability may mean something to the surgeon, but isn’t absolutely clear to a patient. Suppose a certain surgeon is scheduled for 1,000 such operations in the coming year and knows that 950 of those will end successfully and 50 will not. His patient, however, is not scheduled for several hundred operations, and these odds of success hold a different kind of interest for him. He’ll be operated on just this once, and the operation will either succeed or not. Nevertheless, we’d be wrong to say, of course, that his chances are 50-50: they are 95 per cent for him too. But what does that mean exactly?

  Let’s suppose now that our surgeon is a very famous physician who charges $70,000 for his services. But the patient has a choice: he knows of another physician whose success rate – as always, based only on his achievements so far – is 90 per cent, and his fee would be covered by the patient’s insurance plan and would cost him nothing. Which surgeon would you choose? And what if the insurance-sponsored surgeon’s success rates were only 17 per cent? What then? Where would you draw the line?

  Christiaan Barnard (1922–2001) was the South African surgeon who successfully conducted the first ever human heart transplant. When he met Barnard, Louis Washkansky, the patient who was scheduled for that first ever heart transplant, asked the doctor about the chances that the operation would end well for him. Barnard didn’t hesitate and immediately said, ‘80 per cent’. What did Barnard mean by that? I mean, it was the first time in the history of humanity that a human heart was to be planted in the chest of a living patient. No such operation had ever been performed before: it was unprecedented. There were no past operations to compare this one with, and there was no track record to speak of, so what did Christiaan Barnard’s confident statement mean?

  Like most human beings, physicians too often misunderstand the concept of probabilities (except that it’s more dangerous in their case). In his 1992 book Irrationality the British psychologist and writer Stuart Sutherland mentions a study conducted in the United States in which physicians were presented with the following hypothesis. A certain test is supposed to uncover a specific illness. If the tested person is ill, the probability that the test will reveal this is 92 per cent, or 0.92. Then the physicians were asked, ‘What is the probability that the patient does indeed have this illness, given that the test for it proves positive?’ The amazing thing was – at least for people who understand mathematics – that the physicians didn’t understand that these are two completely different things. They thought that the probability of the patient being ill, given the positive result, is 92 per cent too. (Versions of this question appear in many textbooks on probabilities for students of sciences – not to mention the odd fact that a ‘positive’ result in medical terms means that you’re sick.)

  Here’s a simple example that explains the mistake these physicians made. The probability of me taking an umbrella with me when I leave the house after I realize that it’s raining outside is 100 per cent. Yet the probability of rain falling if I’m seen taking an umbrella with me is nowhere near 100 per cent. These are two completely different things – with two completely different probabilities. Similarly, if a person is ill, there’s a 92 per cent chance that the test will reveal that. Yet the chance of the tested person being ill if the test comes out positive is completely different. Suppose the test is for a very scary illness: should the person whose result was positive start panicking at once? Not at all. If we want to know the precise probability of him having this illness, we need more data. For example, we need to know the size of the population segment that contracted the illness, and the percentage of false positives – a situation whereby the test shows healthy people as ill.

  To understand how often the illness scenario is sometimes not even close to 92 per cent likely, here’s a simple example. Suppose only 1 per cent of the population have this particular disease, and suppose the test yields 1 per cent false positives (one of 100 people tested is wrongly diagnosed as having that disease). Let’s further assume that, for the sake of simplicity, 100 people are tested and one of them is ill. Furthermore, let’s be more generous than the authors of the previous case and assume that the test definitely uncovers that one sick person, and that in addition there’ll be one false positive of the remaining 99 tests. In short, two of the 100 were found ill, but only one of them is. Thus, when the test comes back positive, the probability that those tested are truly ill is exactly 50 per cent(!), which is indeed not even close to 92 per cent.

  When a physician comes up with a wrong diagnosis, the consequences might be dire. Shouldn’t physicians, judges and other people who can impact on our lives learn how to think correctly in probabilities?

  LIE DETECTORS

  While you digest that, let me bring up another similar example. Suppose the FBI decide to find out once and for all who really killed J F Kennedy. After years of investigations and leaving nothing to chance, the diligent agents come up with a complete list of all the possible suspects. There are one million people to question (I’ve rounded the number a little) – all of whom are to be given a lie-detector test. Now, let’s assume that when people lie, the lie-detector uncovers their dishonesty 98 per cent of times, but also produces 5 per cent false positives (wrongly indicating that honest people are lying). Now, let’s assume that all 1 million suspects deny any involvement in the Kennedy assassination.

  Out of respect to the inventors of the polygraph, let me say that when the actual murderer is questioned, the machine signals that he’s lying. So what? The machine does the same for another 50,000 people (sadly, 5 per cent of a million is 50,000), and now we have 50,001 positives. It seems that we’re looking at a gang murder. The odds of finding the one person who did the deed among these suspects are 1:50,001.

  I suppose that you’re beginning to understand why uncertain tests that attempt to pinpoint a single incident (a disease that affects one in every thousand, or a single killer in a million) are rather problematic. The surprising results are known as a ‘false-positive-puzzle’. Indeed, our test provided ‘almost certain’ results, but when that ‘almost’ is combined with the rarity of the event tested, we obtain surprising results.

  The conclusion is clear. If a test does not yield absolutely certain results, it is ineffective in spotting a rare incident.

  Chapter 12

  ON FAIRLY SHARING A BURDEN

  In this chapter I’ll present the fair division Airport Problem, which deals with insights on justice that Game Theory provides. Can justice be justified?

  ELEVATING DISPUTES

  Even the eldest tenants couldn’t recall such a fierc
e dispute in their building. It all started when John, who lived on the top (fourth) floor with his wife and 2-month-old twins, suggested or rather begged that the tenants install an elevator (or lift, if you prefer a British lifestyle) in their townhouse. John also wanted all the tenants to share the cost equally. The dispute erupted when Adrian, who lived alone in a rented apartment on the first, or ground, floor, said he wouldn’t pay a dime (or maybe penny) because he didn’t need an elevator and would never use it. Sarah, who lived on the second floor with her husband James and two cats, stated they’d pitch in, but only as a symbol of support – because James was a proud athlete and would never take the elevator, and she’d use it only for particularly large grocery deliveries. Jane from the third floor argued …