Bayes' Theorem

If A 1, A 2 ,... , An are:

Mutually exclusive events.

A 1 unión A 2 unión... unión A n = S.

And B is another event, then:

Bayes' Theorem Formula


p(A1) is the prior probability.

p(Ai/B) is the posterior probability.


20% of a company's employees are engineers and 20% are economists. 75% of the engineers and 50% of the economists hold a managerial position, while only 20% of non-engineers and non-economists have a similiar position. What is the probability that an employee selected at random will be both an engineer and a manager?

Bayes' Theorem Tree

Bayes' Theorem Calculations

The probability of having an accident in a factory that triggers an alarm is 0.1. The probability of it sounding after the event of an incident is 0.97 and the probability of it sounding after no incident has has occured is 0.02.

In an event where the alarm has been triggered, what is the probability that there has been no accident?

I = Accident occurred.

A = Triggered alarm.

Bayes' Theorem Tree

Bayes' Theorem Solution