The maths of randomness: symmetry

Despite the fact that randomness is surprisingly hard to define, we do have a well defined way of describing randomness mathematically with probability theory. (You can read more in the previous article.) And there are two guiding principles in understanding probabilities: symmetry and universality.

Symmetry comes into play in probability theory in the following way: if different outcomes are equivalent they should have the same probability. Two outcomes might look different to you (say a coin landing on heads versus it landing on tails), but the process that produces the outcomes (the mechanics of the spinning coin) is entirely indifferent to which occurs. (You can read more in Struggling with chance.)

This symmetry leads us to judge the probability of a coin landing tails should be the same as it landing heads: giving them equal probabilities of 1/2. Similarly the probability of rolling any of the six numbers on a die should be the same: so each has a probability of 1/6.

Symmetry is a powerful guiding principle in probability theory, as it is in many areas of mathematics. But you have to be careful, even in simple situations, when using it to apply probabilities to the real world.

Two envelopes

For example, imagine you have two envelopes. They both contain a cheque, one for twice as much money as the other. You choose one envelope and look inside and see it contains a cheque for a certain amount. Now you have one chance to decide whether to keep that money, or switch envelopes. What should you do?

Write $x$ for the amount that's in your chosen envelope. This means that the amount of money in the other envelope is either $2x$ or $x/2$. The probability that it's $2x$ is $1/2$ and so is the probability that it's $x/2$. So the expected amount you'll get is $$ \frac{1}{2}\left(2x+\frac{x}{2}\right) =x+\frac{x}{4} = \frac{5x}{4}. $$ Since that's bigger than $x$, you should always swap envelopes. But what if you'd been asked to choose before you looked inside the first envelope? You would have changed your mind, then changed your mind again – obviously something silly is going on!

This isn't really a paradox once you realise that the different possible outcomes are not symmetric. If the cheque in your first envelope said £100,000 – you'd keep it as the threat of losing £50,000 might seem too great. But you might be happy to gamble if you saw £5 in your first envelope.

More importantly, you have to include your prior beliefs of how much the person handing you the two envelopes would be willing to part with. Symmetry suggests that all possible amounts are equally likely which is not only entirely unrealistic, but also leads to an ill-posed probabilistic model. The situation as described above doesn't have enough information to build a complete model. (You can read a full explanation of this famous paradox in The two envelope problem solved.)

This is a trivial and unlikely example, but the point it makes is important. Blindly following the principle of symmetry could lead you to the wrong mathematical model, or indeed to trick you into wrongly thinking you have enough information to create such a model at all. And the consequence of using the wrong mathematical model can be dramatic.

The danger of flawed statistics

In 1999 Sally Clark was tried in court over the murder of two of her children. The defence argued that both children died of Sudden Infant Death Syndrome (SIDS or "cot death"). An expert witness for the prosecution (who has since been discredited) argued that the probability of two children dying of SIDS was the square of probability of one child dying of SIDS – leading to a value of 1 in 73 million for the frequency of two cases of SIDS in such a family. But this argument assumes two such deaths had been independent, whereas it is highly likely that there are unknown environmental or genetic factors that might predispose a family to SIDS, making a second death much more likely.

This incorrect mathematical model, alongside basic errors in the presentation of the statistical evidence, lead to Clark being jailed. Her conviction was eventually overturned on appeal but the experience had a terrible impact on Clark and her family. It serves as an example of the dangers of using flawed statistical arguments. You really have to be very careful, even in simple situations, when applying probability to real world situations.

But probability theory does allow us to work and describe situations where we don't have complete knowledge. Linking this mathematics to real events can be problematic, but trying to understand such real world situations has stimulated development in the maths. And the mathematical description of randomness has allowed us to gain a deeper understanding of the world around us.

Read about the other guiding principle of probability theory – universality.

About this article

Martin Hairer

Martin Hairer is a Professor of Pure Mathematics at Imperial College London. His research is in probability and stochastic analysis and he was awarded the Fields Medal in 2014. This article is based on his lecture at the Heidelberg Laureate Forum in September 2017.