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A Quick Guide to Bayes’ Theorem
Mr Smith has two children. At least one of them is a girl. What is the probability that both children are girls?
If you said 1/2 then you are not correct! As you will see, the actual answer is 2/3.
Bayes’ theorem describes the probability of something happening based on prior information about the event. It can also be interpreted as an approach for expressing the degree to which one believes something based on available evidence.
The following is a brief introduction to Bayes’ theorem that — be warned — uses some mathematical notation. However, it’s written to hopefully be accessible even with very little math knowledge.
The Turkish Van cat problem
Let’s say that you spot a cat with completely white fur, except for an orange tail and ears. At once, you realise you might have stumbled across the rare breed of Turkish Van cat patrolling the neighbourhood. However, not being satisfied with just thinking you’ve seen this rare breed of cat, you want to know how sure you can be.
In fact, you want to know the probability that you have just seen a Turkish Van cat given your observation of a cat with a ginger tail and ears. The way we write this probability is as follows:
P(Turkish Van cat | Ginger tail and ears)
In other words, the probability (P) of seeing a Turkish Van cat given (|) you have seen a cat with ginger tail and ears.
From a quick Google search, you know that 95% of Turkish Van cats have these distinctive orange tails and ears, therefore you can say that,
P(Ginger tail and ears | Turkish Van cat) = 0.95.
Notice how this is the reverse of the above statement. Now we’re saying that given we have a Turkish Van Cat, the probability of it having a ginger tail and ears is 95%. On the other hand, the probability of a non-Turkish Van cat having a ginger tail and ears is only 2%. Hence,
P(Ginger tail and ears | Not a Turkish Van cat) = 0.02.
What other information might we need?
It would be helpful to know how rare Turkish Van cats are. So we search for this information too. It turns out they make up only 0.1% of the…
