Bayes' Theorem

The figure shows the sample space of a random experiment. E1, E2, E3 and E4 and four MECE events.

Consider the event X, which is formed of some elements from each of E1, E2, E3 and E4.

We can write:

n(X) = n(X∩E₁) + n(X∩E₂) + n(X∩E₃) + n(X∩E₄)  

Dividing throughout by n(S), we get:

P(X) = P(X∩E₁) + P(X∩E₂) + P(X∩E₃) + P(X∩E₄)  

Generalising, P(X) = ∑ P(X∩E_i) = ∑ P(X E_i)

The above result is called the law of total probability

The Law of Total Probability

Bayes’ Theorem works like this: Starting from P(X/E₁), P(X/E₂) etc. Bayes’ theorem helps us to calculate P(E₁/X).

Here is how:

Using the product rule of probability, we may write:

P(E₁ X) = P(E₁/X) . P(X) = P(X/E₁) x P(E₁)

Thus, P(E₁/X) . P(X) = P(X/E₁) x P(E₁)

Hence

P(E₁/X) = P(X/E₁) x P(E₁) / P(X)

              = P(X/E₁) x P(E₁) / ∑ P(XE₁) ...... Bayes' Theorem