8 Feb 2021

As far as I remember, the word “philosophy” means “love of knowledge”. Some of the philosophers whose books were in my college library tried to prove that God knew all the answers and others that truth lay in empirical observation or the meaning of words.

“Whereof we cannot speak, thereof we must pass over in silence” – Wittgenstein.

Somewhere buried in their philosophical texts one might find a grudging reference to probability. John Locke wrote that probability “is to supply our want of knowledge”. In the search for certainty, probability was to some, it seems, as admission of defeat, a last resort.

A brief disclosure: the only thing written by me in the college library are the letters zzzzz carved into the leg of a table. The fact that I couldn’t see what Locke, Descartes and Wittengenstein were so exercised about was confirmed by my examination results. But I value the awareness of probability as highly as anything else.


 Some people point to the fact that humans initially learn by imitation and get hung up on the observation that animals do that too. The ability to observe that, if A, then B, puts animals on the first step of logical thought. When I pick up my dog’s lead she immediately starts to celebrate her forthcoming walk. You could say that she thinks the probability of a walk is 100%. In Locke’s terms, the sound and sight of the lead being picked up has supplied her want of knowledge. 

The weakness in my dog Hattie’s understanding of probability is her failure to appreciate that there are any numbers between 0% and 100%. Her world is essentially binary. But she should not be too despondent. Humans sometimes think in exactly the same way.

The easiest example of probability is 50/50. When we toss a coin we know, assuming no skullduggery, that a head or a tail is equally likely. (Dogs always expect tails, obviously).

We should also know, though gamblers sometimes don’t agree, that no matter how many times the same side comes up in a row, the odds do not change for the next toss. 

For what it’s worth, this feat of sophisticated reasoning sets us well apart from any animal of my acquaintance. Betting on a 50/50 chance is gambling at its purest and lowest. There’s nothing wrong with gambling as such as long as we never fool ourselves that we have special powers to influence an outcome. 


In investment and in many other aspects of life, an appreciation of probability is the foundation of reason. In John Locke’s time, an idea was the beginning of a process of judgement towards a conclusion. He was not constantly bombarded by social media causing him to ask “how likely is this to be true?” a dozen times a day. That’s why he had the luxury of using probability as a last resort rather than a first line of defence.

We are not so lucky. Like the Queen in Lewis Carroll’s Alice in Wonderland, we might be asked to believe six impossible things before breakfast. 

Probability is a mathematical discipline as in the cases of the coin toss or the more difficult Monty Hall problem where a game show contestant is trying to find a car behind three doors. But it goes well beyond mathematics. Before we make our calculation we need to ask if we are starting with the right numbers. 


Much probability is based on empirical experience and by implication repetition. A hack golfer like me is much less likely than a professional to hole a ten foot putt. The probability that I will sink my putt and he will miss his is slight – but it is not zero, despite what people who have seen me playing golf might say. 

The chances of the four outcomes – a) he succeeds and I fail; b) we both miss; c) we both succeed; d) I succeed and he fails – should probably be ranked in that order but the odds attached to each are matters of judgement. 

We will apply mathematics to come up with probabilities by making some assumptions for the sake of the exercise. I will assume that the professional will hole a ten foot putt 55% of the time and, generously, that I will do so on 5% of my attempts. On that basis a) will happen 52.25% of the time, b) 42.75%, c) 2.75% and d) 2.25%. If I am satisfied with my estimates of the basic assumptions (based on experience and observation) I know that if someone offered me odds of 50-1 against d) I should in theory accept the bet.

And this illustrates a difficulty that besets us all constantly. The chill verdict of probability is difficult to accept in the case of outcomes that are either overwhelmingly likely or very unlikely. 


There are two trapdoors to financial ruin. One is confusing improbability with impossibility. The so-called Black Swan event. These rare creatures suddenly appeared in armadas during the financial crash just over ten years ago. One week Northern Rock was offering great rates to its savers, the next its customers were grimly queuing outside its Moorgate branch (just next to my office at the time) trying to withdraw all their money. 

The other trapdoor is sprung by the opposite mistake – betting too enthusiastically on an unlikely outcome due to the tendency to believe that really wanting something to happen makes it more probable. Everyone loves a good narrative and we are inclined to select facts that support the story and dismiss those that do not. This is fertile ground for fraudsters. The few whistleblowers who warned that Madoff’s results were too good to believe discovered that for most people too good was a positive reason to believe. Keeping the faith in anything can feel awfully virtuous. 

In retrospect we can say that both the credit rating agencies who marked bundles of mortgage backed securities as AAA in 2008 and the high profile investors who begged to be allowed to invest with Madoff had lost their reason to some extent. In these cases we might attribute their mistakes to greed or even fear – the idea that they were wrong was just too awful to contemplate.

I say that probability is the root from which reason flourishes. It is critical in making sensible investment decisions. It takes us through a process that asks not what is this asset theoretically worth but is today’s price of this asset compatible with what I know of the world? UK government paper currently pays a yield of 0.5% and BP shares offer a dividend yield of 10%. Are those two risks sensibly priced relative to each other? There is no correct answer beyond our judgement of probability. 


In any given year it appears that my chance of being diagnosed with cancer, according to my age range and gender, is 0.24% (source internet search and back of envelope).

What were my chances of dying of Covid-19 in 2020? Assuming, rather arbitrarily, that the official number of cases (4 million) understates the truth by a factor of five, then 20 million in the UK have had it. That means each member of the 67 million population had a 30% chance of catching it, though in many cases being asymptomatic and unaware. The Covid fatality rate for my age range, with no comorbidities (thank you for introducing us to that word, coronavirus) appears to be around 1.1%. Crudely, my chances of dying from the virus were about 0.33% (30% x 1.1%). 

Translate these statistics to a news headline and you could say “CROW 50% MORE AT RISK FROM COVID THAN CANCER”. But reason kicks in and points out that these are both real but very small risks. They are black swans and while it  would be a mistake to act as if they were impossible, only fear could make make us behave as if they were of pressing importance. Reasonable precautions should be taken but please don’t close every school in the country and bankrupt the catering industry on my account. 

Rational beings should never hesitate to test the probability of any doubtful or contentious proposition, however confidently asserted. Dogs are loved for their trust and fidelity but we are blessed with the cynicism otherwise known as reason.

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