Imagine there is a terrible disease reported, and although it affects only one in ten thousand people, it is absolutely lethal. You are worried about it, so you decide to undergo a medical test to see if you have the disease. Now, no medical test is ever 100 per cent accurate, but your doctor explains that this one is known to be 99 per cent accurate, regardless of whether or not you have the disease (in other words, it will deliver a correct positive or negative result 99 per cent of the time). You decide to take the test. You’re a little nervous, but you think it’s a sensible thing to do. A blood sample is taken, and you’re told the results will be sent to you in the post.
A week later the envelope arrives from the testing centre. You open it up, and read the contents. Staring you in the face is the answer you dreaded: the results are positive. The test has indicated that you have the lethal disease. You are devastated.
And you are right to be, aren’t you?
Just for a moment, review the scenario above and ask yourself what may seem like a very easy question: how likely are you to have the disease? Please decide on an answer before reading any further. If you are unsure, just settle on an estimate.
I imagine most of you have re-read the above and confirmed that the answer must be that you are 99 per cent likely to have the disease. Those of you made suspicious by my request to reconsider your answer may have been nervous about committing to that percentage, but would certainly commit to saying that you are far more likely than not to have the disease. Only a few of you, trained in statistics, will know the correct answer. From the information given above, you should not be concerned about the positive test results. You are less than 1 per cent likely to have the disease.
That’s right – although the test is 99 per cent accurate, you are less than 1 per cent likely to have the disease. The key to unlocking this counter-intuitive fact lies in one piece of information you may have missed, or at least not properly factored in: the disease hits only one in ten thousand people. So your positive test result, with its 99 per cent accuracy, could mean that you are one of the 99 per cent of people who have been correctly told that they have the disease, or you could be one of the 1 per cent of people who don’t have the disease but who have been wrongly told that they do. (Remember that the test is 99 per cent accurate, regardless of whether or not you have the disease.) So, are you more likely to be one of the correctly diagnosed people with the disease, or one of the wrongly diagnosed people without it?
The disease strikes only one in ten thousand. So, forgetting the test for a moment, you can immediately see that you are far more likely (to the tune of 9,999:1) not to have the disease. Now, let’s imagine that a million people take the test. Only a hundred or so of those will actually have the disease. Ninety-nine out of those hundred will be correctly diagnosed as having it, because the test is 99 per cent accurate. On the other hand, 999,900 people won’t have the disease, but 1 per cent (or 9,999 of them) will be wrongly diagnosed as having it. So are you one of those ninety-nine who have it, or one of those 9,999 who don’t? You’re over one hundred times more likely to be in the second, safe, category.
The last fifteen years or so have seen a genuine attempt by cognitive scientists to research our fascinating tendency to fall prey to cognitive illusions, which are mental traps as persistent as optical illusions. The problem above is not just a clever puzzle, it is a potentially devastating real-life possibility and speaks volumes about our inability to understand probability and risk.
I toss a coin seven times, and record whether it lands heads (H) or tails (T). For my own perverse enjoyment, I write down the following list and give it to you. It shows three results, but only one of them is the real outcome. Which is most likely to be the real result?
1. HHHHHHH 2. TTTTTTH 3. HTTHTHH
Make a mental note of your answer. Be honest.
The answer is that each combination is as likely as any other. A coin is no more likely to land on any one specific combination of heads and tails than it is to land on any other, including the same way up every time. This is because the coin has no memory; each throw will yield a fifty-fifty result of H or T. However, we are seduced by the fact that the third option looks more ‘typical’. In the same way, who would dream of choosing the following lottery numbers: 1, 2, 3, 4, 5 and 6? Yet the chance of this sequence being the winning combination is as likely as any less obvious sequence you may choose.
This confusion of ‘probable’ and ‘typical’ leads to the famous ‘gambler’s fallacy’. I have sat at the roulette wheel and watched suited, impressive, Chinese pros meticulously note down the results of wheel spins to decide which colour is most likely to come up next. Clearly, if the ball has landed in black five times in a row, it would seem sensible to bet on red. Simply not true! While it is the case that in an infinite number of spins the colours would be expected to even out, the colours may only roughly balance over a very large number, and certainly are not required to balance over a short number of spins.
Another common obstacle stands in the way of our rationality. The cognitive researcher Massimo Piattelli-Palmarini offered the following game. Firstly, give yourself five seconds to come up with a rough answer to the following multiplication in your head. You won’t have enough time to work out a proper answer, so hazard a guess:
2 x 3 x 4 x 5 x 6 x 7 x 8
Please note your answer, and have a few other people try it too. Note their answers as well.
Next, find some more people of the same intellectual level and have them offer an answer to the following, with the same restriction of five seconds:
8 x 7 x 6 x 5 x 4 x 3 x 2
As Piattelli-Palmarini perspicaciously points out, the second set of results will be significantly greater than the first. How can that be? Looking at both, we know that the answers must be the same, but somehow the second sequence yields higher estimates.
That’s not all, though. Compare the estimates, your own and those of your friends, to the real answer. The correct answer is . . . 40,320. Did you find your estimates were much, much lower than the truth? The answer lies in the fact that we begin to multiply the numbers (2 x 3 x 4 . . .) and then, once we have an estimate at this point, we find it very hard to stray far enough from it. It’s enormously difficult to let numbers boldly multiply in our heads. We root ourselves somehow to figures we have in our minds, and get stuck. It’s even been shown that if a trial lawyer suggests a sentence of a number of months for a criminal, the judge is very likely to think close to that figure when arriving at his decision, even though he may appear to disregard it. Where suggested numbers are higher, judges return with higher sentences themselves. It’s as if we set ourselves a yardstick once we hear or see a number and can’t stray very far from it. This is a common cognitive pitfall known as anchoring. If an enormous sheet of newspaper was folded over on itself a hundred times (let’s imagine that’s possible), how thick would the eventual wad of paper be? The thickness of a brick? A shoe box? As Sam Harris points out, to prove a similar point about intuition in his excellent book The End of Faith, the correct answer is that the resulting object would be ‘as thick as the known universe’. Again, we anchor ourselves to the initial smaller measurements.
proudly affirmative answer. Yet we are somehow wired to make these sorts of mistakes, despite the fact that we know better. It’s almost as if we know certain information, but just can’t bring ourselves to use it.
Now, consider the following:
Harry was very creative as a child and loved attention. He didn’t always feel ‘part of the gang’ and this led to a desire to try to impress others with his talents. He went through school rather self-obsessed, and tried his hand at any creative field. He really enjoyed any opportunity to give a presentation or to show off in front of an audience.
Take a look at the following statements regarding Harry as an adult, and place them in order of most likely to least likely:
1 : Harry is an accountant.
2 : Harry is a professional actor.
3 : Harry enjoys going to classical concerts.
4 : Harry is a professional actor and enjoys going to classical concerts.
Please go ahead and mark them in the order you think is appropriate, before reading any further.
Did you decide Harry was more likely to be an actor than an accountant? Mistake number one. There are many, many more accountants in the UK than professional actors. Can accountants not be self-assured and good speakers? Because the description fitted your sense of how a ‘typical’ actor might describe his background, you were blinded to making a sensible estimate.
Did you decide that number four was more likely than number three? Stop and think: how can ‘Harry is a professional actor and enjoys going to classical concerts’ ever be more likely than just ‘Harry enjoys going to classical concerts’? In number three, Harry could do anything as a job. The probability of two pieces of information being true is necessarily lower than that of just one of them being true. It sounds obvious now, but we still tend to blindly choose the necessarily less likely option as the more likely.
Imagine I have snuck a loaded die (I use the correct and irritating singular of the word ‘dice’) into one of Las Vegas’s fine casini. The die is cleverly weighted to provide a bias towards rolling a six. I would expect it to roll a six most of the time, but not all. I roll it a few times and log the results. Decide which of the following is more likely:
1. 3, 1, 6, 2, 5 2. 6, 3, 1, 6, 2, 5
By now you are catching on, but it’s so tempting to say the second option, isn’t it? Yet the second option is necessarily less likely than the first, for the reasons given in the previous example. It is, you will note, the same as the first sequence, plus the extra condition of rolling a six at the start. It can only be less likely, as in the first example, when any number could be rolled first. If you prefer, you can remove the six from the start of the second result and place it at the end, so the two lines look more similar – you will still be tempted to say that the second result is more likely.
We are hopeless!
If you really want to start a fight, may I suggest the following classic example of counter-intuition? Known now as the ‘Monty Hall Problem’, its first appearance was probably the one presented in Bertrand’s Calcul des probabilités (1889), where it was known as Bertrand’s Box Paradox. Famously, it was presented to the ‘Ask Marilyn’ advice column in Parade magazine in 1990, and the answer given in
the column, though correct, sparked a storm of controversy. Brilliantly, Marilyn proved a lot of mathematicians wrong. Here is my version of the problem, as featured on TV, with which I have caused many bitter arguments:
I show you three ring boxes and one very expensive ring which you are trying to win. While your back is turned, I place the ring into one of the ring boxes. I know which one it goes into, but you don’t. Then you turn back round and I explain the game. I will ask you to guess in which box I have placed the ring. You can point to any box of the three to make your choice. Then, before asking you to commit, I will open one of the remaining boxes which I know does not contain the ring. I’ll show you that box is empty, and remove it from the game. Then I shall ask you whether you want to stick with the box you have chosen, or whether you would rather switch to the final, remaining box. If you choose the correct box, you keep the ring.
Should you stick, or switch?
What would you do? Do you gain anything by switching? People have very different answers to the problem, but the most common is that they would stick. Somehow it feels right. After all, if it’s only fifty- fifty at the end, why switch? Surely it’s more frustrating to change and then find you were right the first time?
The answer is, you should always switch. You are always twice as likely to win the ring if you do so. In fact, the chances increase from one-third to two-thirds by changing. It isn’t fifty-fifty.
If this seems ludicrous or just plain wrong to you, let me explain in a few different ways, so that you can pick the explanation you can most easily follow. Firstly, rest assured that this stumps all sorts of bright, intelligent people and even mathematicians, if they don’t know it.
Look at it like this. Let’s say I will always put the ring in box C. ‘C’ is our winning box. Two out of three times, you’ll first choose A or B and be wrong. That means that two out of three times, if you think about it, I have no choice which of the remaining two to open and remove, as one of the two boxes you leave for me to choose between will be the winning box C. If you choose A, I must open B and leave C, and if you choose B, I must open A and leave C. In both cases, I am avoiding the winning box. In both cases, you should switch to the one I avoid because it will be the winning box. On the rarer occasion (one out of three) when you unknowingly pick the winning box C straight away, I can then open either A or B, and of course you shouldn’t switch. But that’s only in one out of three cases, the time you happen to get it right on the first go. Please read that again.
If you still want to see those final two boxes as fifty-fifty, imagine it this way. Let’s pretend there were a hundred boxes, not three. When you make your first choice, it clearly has a one in a hundred chance of being right – very unlikely. Now – and imagine this is happening – I reveal as empty ninety-eight other boxes, leaving yours and one other one (in, say, the thirty-seventh position) closed. Now, bearing in mind I know which box has the ring in it, isn’t it much more likely that the one I’ve chosen to leave closed (number thirty-seven) contains the ring, rather than the one you chose in a one in a hundred chance? The two boxes don’t look fifty-fifty now, do they? Your first choice is still very unlikely, and number thirty- seven seems much more probable. That’s because your first choice in this example would have a 1 per cent (one in a hundred) chance of being right, which means the only other possibility (the final box) must necessarily have a 99 per cent chance of being right. The two have to add up to a 100 per cent chance, as we know it’s in one of them. Similarly, in the three-box game, your first choice has a one-third chance of containing the ring, quite naturally, and that means that the final box I leave for you must have a two-thirds chance.
You are assured that the answer is exactly as explained. Enjoy the fruits of bringing the problem up in company.
You might argue that this last example is an interesting thought-exercise, or even a theme for a bewilderingly successful game show, but rather divorced from real life. The fact remains that cognitive traps make us unwittingly prone to drastic misunderstandings of probability, and this can undoubtedly lead people, including doctors and jurors, to make terrible decisions.
In an earlier chapter, you learned that the way in which you view images in your head makes a huge difference to the extent to which you can connect with them. That leads to one of the biggest causes of the sort of irrational thinking we are discussing. If something can be easily pictured, it feels more real and immediate than something that does not convert itself into such a powerful picture. Somehow the rolls of the loaded die which included a couple of sixes looked more real, and we felt it as more likely even though we knew they couldn’t be. Equally, a drug that promises to reduce the mortality rate of a disease from 10 to 0 per cent seems much more worthy of investment than one that reduces the mortality rate of a different disease from 40 to 30 per cent, even though each will do as much good as the other. We look at pictures and read reports of rail crashes and play out terrible sequences in our heads, deciding it’s dangerous to travel, yet we are twice as likely to have an aeroplane crash into our house (1 in 250,000) than we are to die in a rail accident (1 in 500,000).* We are in an area where emotion rules over reason, and the results can be damaging.