Venn diagram showing the logical structure of the appearance of spots and measles
This tells Dr Why that there are x patients who have measles and spots, m
patients who have measles, while the total number of patients overall is N. From
the diagram he can see that the probability that someone has measles and has
spots is simply x/N, while the probability that someone has measles is m/N. The
conditional probability, the probability that someone has spots given that they
have measles, written prob(S|M), is x/m. Putting these together, Dr Why gets the
probability that someone has both measles and spots
or
prob(M & S) = prob(S|M) × prob(M)
and similarly
prob(M & S) = prob(M|S) × prob(S)Bayes’s formulaBayes’s formula
Equating the expressions for prob(M & S) gives Bayes’s formula, the
relationship between the conditional probability and its inverse. Dr Why will have
a good idea about prob(S|M), the probability that if a patient has measles, they
have spots. The conditional probability the other way around is what Dr Why is
really interested in, his estimate of whether a patient has measles if they present
with spots. Finding this out is the inverse problem and the kind of problem