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 n1is the outcome for event A, and n2for event B, then:
P(A) = n1/n, P(B) = n2/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 n1 and n2 are completely different from each other. When calculating total outcomes for events either A or B is n1+n2.
P(A + B) = n1 + n2/ n = n1/n + n2/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
