Probability — Part 2: Total probability, Bayes' theorem and random variables: visual lesson, ISC Class 12 Mathematics

Lesson

Probability, Part 2

From causes to evidence, and back again.

In this lesson you will

Find the probability of an event that can happen through several mutually exclusive causes
Apply Bayes' theorem to find the probability of a cause given the evidence
Construct the probability distribution of a discrete random variable
Calculate and interpret the mean of a random variable

Part 2 at a glance.

Diagram

$A$ is cut into the pieces $A \cap E_1$, $A \cap E_2$, $A \cap E_3$.

Causes that do not overlap and together cover everything.

Concept

$P(A) = P(E_1)\,P(A \mid E_1) + \cdots + P(E_n)\,P(A \mid E_n)$

Total probability: add prior times likelihood.
Step	What to write
1	Name the causes $E_1, \ldots, E_n$ (a partition)
2	Write the priors $P(E_i)$
3	Write the likelihoods $P(A \mid E_i)$
4	$P(A) = \sum P(E_i)\,P(A \mid E_i)$