Theorem of total probability will present independent event, suppose event A and event B. There are two specific situations for this theory to prove:

- If two events A and B are mutually exclusive and
- If the two events A and B are not mutually exclusive.

We’ll explain them one by one:

**Firstly,** if those two events are mutually exclusive then the occurrence of either A or B is

P (A + B) = P (A) + P (B)

Let us considern for all possible outcomes in an experiment which are mutually exclusive. If n_{1}is the outcome for event A, and n_{2}for event B, then:

P(A) = n_{1}/n, P(B) = n_{2}/n

For this first experiment we have taken events A and B are mutually exclusive, so we’ll definitely point out that the two outcomes n_{1} and n_{2 }are completely different from each other. When calculating total outcomes for events either A or B is n_{1}+*n _{2}.*

*P(A + B) = n _{1} + n_{2}/ n = n_{1}/n + n_{2}/n = P(A) + P(B)*

**Secondly,** if the two events A and B are not mutually exclusive, then probability of occurrence of events A and B is

P (A +B) = P (A) + P (B) – P (AB)

**Note:** Here the event A + B means the occurrence of one of the following mutually exclusive events: AB, AB and AB. Hence,

P (A +B) = P (AB + AB + AB) = P (AB) + P (AB) + P (AB)

Then it’s:

P (A) = P (AB) + P (AB)

=> P (AB) = P (A) – P (AB)

And P (B) = P (AB) + P (AB)

=> P (AB) = P (B) – P (AB)

So we get:

P(A + B)= P(AB) +[P(A)- P(AB)]+[P(B)- P(AB)]

= P (A) + P (B) – P (AB)

If for example, events A, B and C are not mutually exclusive, then we’ll use:

P (A + B+ C) = P (A) + P (B) + P(C) – P (AB)-P (AC) – P (BC) + P (ABC)

To go further into the theorem of total probability we have to memorize two other equations:

**Boole’s inequality**: Here we’ll consider P (A +B) ≤ P (A) + P (B). One thing must be noted here. Since we are placing P (AB) = 0, we are trying to provide information on even A and B that are mutually exclusive.**Bonferroni’s inequality**: Here we’ll consider P (AB) ≥ P (A) + P (B) – I

**Links of Previous Main Topic:-**

- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution

**Links of Next Statistics Topics:-**

- Random variable
- Binomial distribution
- What is sampling
- Estimation
- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory