Do we really understand numbers? By Andrew Tottenham, Managing Director, Tottenham & Co March 17, 2020 at 8:07 pm In the light of the COVID-19 pandemic, I thought I would write about numbers. Most people think they understand numbers and how they work. As children, we are taught to add, subtract multiply and divide. Some go on to study more advanced mathematics, but as a whole, humans are not very good about understanding numbers. Certainly, we know that ten is double five and a piece of wood five centimetres long is half of a piece of wood ten centimetres long. We can even measure a line and imagine reasonably accurately what double or even three times the size might look like. Our brains are fairly good at figuring out linear problems. We are not bad with volumes either; we can look at a glass with some water in it and estimate what double the amount of water would like. We’d probably just estimate in one dimension, how much higher up the side of the glass would the surface of the water be, but wouldn’t be so good at determining how much water there would be if each dimension were doubled. We are also not particularly great at understanding random sequences. Humans are remarkably good at spotting patterns; it is believed that this is what makes us excellent at recognising familiar faces in a crowd and may be an evolutionary benefit of being able to spot well-camouflaged wild animals hiding in the grass when we roamed the African plains. Faced with a series of numbers, many people will try to “make sense” of them, consciously or subconsciously, looking to see if there are underlying patterns in the series. If a random number generator produced a series of ten even numbers in a row, the odds of the next one being even are exactly the same, 50:50, provided it is a true random number generator. And yet, this is the gambler’s fallacy: Gamblers will watch a roulette wheel and if they see a sequence of ten red or black or odd or even numbers in a row, they immediately start betting on the opposite to occur, with the thought that the odds of the opposite happening must be higher than before. Roulette wheels, like random number generators, have no memories. One event is not linked to the next. Understanding probabilities is something most of us are really not very good at. A good example is the “Let’s Make A Deal” problem. “Let’s Make A Deal” was a game show hosted by Monty Hall that ran for many years on American television. At the end of the show, the winning participant was given a choice of three doors, behind one of which was something of high value, a new car, perhaps. Behind the other two were items of a much lesser value; one might be a sofa, for example. They would then open the door and whatever was behind the door the participant would win. Now to show how fallible we are, I will give you a famous problem in probability. In this example, the participant makes the choice, but the door is not opened. Instead, Monty opens one of the other two doors to reveal a sofa and gives the participant the option of changing their choice and pick the other closed door. What should they do? Should they change and pick the other remaining closed door or should they stay with their original pick? There is only one car and one sofa and only two doors so the odds are 50:50 of picking the car; changing the selection shouldn’t make a difference. And yet it does. Switching doors doubles the chances of winning the car! If they switch doors, the participant’s odds go from one in three to two in three. If you don’t believe me, you’re in good company. When this problem was published in a column written by Marilyn vos Savant in Parade magazine in 1990 she gave the answer that switching doors doubled the chances of winning and was castigated for it. Mathematicians wrote into the magazine and in other publications claiming that she “blew it”, that she “didn´t know what she was talking about”. Even the eminent number and probability theorist, Paul Erdős, signed his name to an open letter criticising Marilyn for her error. And yet she was right. It took a computer simulation of the problem before Erdős believed it. I won´t show you why, but I will in my next column. The other thing humans are not at all good at is understanding exponentiality and this is why, in my opinion, our politicians have mainly failed us with dealing with the novel coronavirus which causes COVID-19. Reacting early and positively is the key. What has been shown in China and Italy is that without “social distancing”, the number of infected people doubled approximately every three days. We also know it can take up to 14 days for the symptoms to show, so when we are looking at the number of cases, this is in fact from viruses being transmitted up to 14 days previous. Also, this is likely to be well-understated due to the fact that many, perhaps 80%, have minor or no symptoms. As of today, London has a reported number of 407 cases of COVID-19. As I said, it is likely very much higher, but let’s use the number of 400 cases. If there were no “social distancing”, which has been found to slow the transmission of the virus, how many cases would there be by mid-April? Thirty days is ten lots of three days; the number of people infected would double ten times (210) or 400 multiplied by 1,024 or 410,000 approximately. After another three days, the number would be 820,000 people infected and after another three days … As rapidly as exponential growth gets out of hand, slowing the infection rate can have just as a dramatic impact in the other direction. If the rate were reduced to doubling every 15 days, in mid-April there would only be 800 cases. Quite a difference. That is the nature of exponential, as opposed to linear, growth and unfortunately, our brains tend to think linearly as opposed to exponentially. Luckily, we have many really smart people devoted to working on the problem and I think it will get “solved”, but I don’t know when. Until then, we as individuals can only take actions to slow down the infection rate. In the exponential world small actions can have big impacts.