The Game of Chance
Last week’s blog featured a book called ‘Maths on the Back of an Envelope’ by Rob Eastaway, which might make a useful read for budding mathematicians over the summer. There were just a couple of points that I wanted to finish off before we move on, both related to some of the difficult decisions we need to make as we assess where our money goes. Eastaway poses a question about what uses the most energy around the house. Is it the fridge, the television on standby, the shower or the kettle? The only certain answer, he tells us, is that it is not the television on standby. The others will depend on lots of variables, and any one on a given day could be the biggest – which is not entirely helpful if we are looking to make savings, but which makes the point that there are no simple answers.
Likewise, we are warned against the dangers of taking numbers for granted, particularly big numbers, which we are notoriously poor at grasping. It should not be too hard to work out whether or not forty new hospitals have been built, though my guess would be that we will be arguing about what is a hospital and what is not for years to come. However, Eastaway asks the useful question that when the government announces a £1 billion increase in health spending, is that significant? If we assume that there are approximately 50 million people in England, that works out at only £20 per person, which is not going to change very much – though, of course, it won’t be exactly £1 billion and it won’t be spread evenly between 50 million people, so in some areas it could prove significant.
Another book about numbers that might be a good summer read is Leonard Mlodinow’s ‘The Drunkard’s Walk – How Randomness Rules Our Lives’, which I found an interesting read, with the sort of eclectic knowledge that floats my boat. For example, we are told that while the Greeks produced mathematical luminaries like Archimedes, Euclid and Pythagoras, the Romans did not produce a single mathematician of any note. In Roman culture it was comfort and war, not truth and beauty, that occupied centre stage. But while they found little value in abstract geometry, their practical nature gave them a considerable focus on probability, which Cicero called ‘the very guide of life’.
A challenge that might be familiar is the so-called ‘birthday problem’, which consists of asking how many people a group must contain in order for there to be a better than even chance that two members of the group will share the same birthday – assuming all birthdates are equally probable. Most people think the answer is half the number of days in a year, or about 183. But that is the correct answer to a different question: how many people with different birthdays do you need for there to be a better than even chance that one of them with share your birthday? If there is no restriction on which two people will share a birthday, the fact that there are many possible pairs of individuals who might have shared birthdays changes the answer drastically. In fact, the answer is astonishingly low: just 23.
Mlodinow offers what he calls another crazy game. Suppose the state of California made its citizens the following offer: Of all those who pay the dollar or two to enter, most will receive nothing, one person will receive a fortune and one person will die in a violent manner. Would anyone enrol in that game? People do, and with enthusiasm. It is called the state lottery. And although the state does not advertise it in the manner in which he has described it, that is the way it works in practice. For while one lucky person wins the grand prize in each game, many millions of other contestants drive to and from their local ticket vendors to purchase their tickets, and some die in accidents along the way. Applying statistics from the National Highway Traffic Safety Administration, and depending on such assumptions as how far each individual drives, how many tickets he or she buys, and how many people are involved in a typical accident, you apparently find that a reasonable estimate of those fatalities is about one death per game.
Perhaps unsurprisingly, the author tells us that people have a very poor conception of randomness. They do not recognise it when they see it and they cannot produce it when they try. What is worse, we routinely misjudge the role of chance in our lives and make decisions that are demonstrably misaligned with our own best interests. And this is nothing new. As the mathematician Jakob Bernoulli put it in the seventeenth century, ‘One should not appraise human action on the basis of its results.’ Or as the philosopher Immanuel Kant wrote in 1784, ‘Each, according to his own inclination follows his own purpose, often in opposition to others; yet each individual and people, as if following some guiding thread, go toward a natural but to each of them unknown goal; all work toward furthering it, even if they would set little store by it if they did know it.’
Developing his point, Mlodinow says that while people may pay lip service to the concept of chance, they behave as though chance events are subject to control. In real life, it is often very difficult to resist the illusion of control. Research has shown that this illusion is enhanced in financial, sports and, especially, business situations when the outcome of a chance task is preceded by a period of strategizing (endless meetings), when performance of the task requires active involvement (long hours at the office), or when competition is present (this never happens, right?). The first step in battling the illusion of control is to be aware of it. But even then, it is difficult because once we think we see a pattern we do not easily let go of our perception.
Looking to the wisdom of the past again, as philosopher Francis Bacon put it in 1620, ‘The human understanding, once it has adopted an opinion, collects any instances that confirm it, and though the contrary instances may be more numerous and more weighty, it either does not notice them or else rejects them, in order that this opinion will remain unshaken.’ Psychologists today call this confirmation bias, when we make up our mind about something very quickly and sometimes spend the rest of our lives trying to justify our decision, even in the light of powerful evidence that we are wrong.
We miss the effects of randomness in life because when we assess the world we tend to see what we expect to see. We in effect define degree of talent by degree of success and then reinforce our feelings of causality by noting the correlation. That is why, although there is sometimes little difference in ability between a wildly successful person and one who is not as successful, there is usually a big difference in how they are viewed.
Moving to his conclusion, Mlodinow explains that the cord that tethers ability to success is both loose and elastic. It is easy to see fine qualities in successful books or to see unpublished manuscripts, inexpensive vodkas or people struggling in any field as somehow lacking. It is easy to believe that ideas that worked well were good ideas, that plans that succeeded were well designed, and that ideas and plans that did not were ill conceived. And, he continues, it is easy to make heroes out of the most successful and to glance with disdain at the least. But ability does not guarantee achievement, nor is achievement proportional to ability. And so it is important to always keep in mind the other term in the equation – the role of chance.
He believes it is important to plan, if we do so with our eyes open. More importantly, we ought to identify and appreciate the good luck that we have and recognise the random events that contribute to our success. Chance events may cause us grief, but we must also be fully aware of our good fortune in the absence of bad luck, the absence of events that might have brought us down, and the absence of disease, war, famine and accident that have not – or have not yet – befallen us.