Total Probability Theorem

Mathematically, probability means the chances of occurrence of an event. The probability of the likelihood of an event can be 0 or 1. Probability has a rule that relates conditional probability to marginal probability. This rule is known as the law of total probability. It divides the complete event into various sub-events so that finding the probability is easier for the complete event.

Law of total probability

Two events A and B, which have a sample space of S, then that sample space can be denoted as A ∩ B′, A ∩ B, A′ ∩ B, A′ ∩ B′. This set of events is known as a mutually disjoint event or pairwise disjoint event because of its disjoint property. Consider the Venn diagram as shown below. We use the law of total probability theorem in those cases where we need to find the probability of the likelihood of an event, and that likelihood depends upon the occurrence of other events.

Law of total probability statement

Consider a series of events C1, C2 … Cn that belongs to a sample space S. All the events have a non-zero occurrence probability. This means these events will occur at any cost. For any event A, which is associated with sample S, the mathematical expression according to the total theorem probability is given by:

Proof of total probability theorem

Example: 0.42 and 0.90 are the probabilities of completion of the job on time with and without rain, respectively. If the probability of occurrence of rain is 0.45, then determine the probability that the job will be completed on time.

Solution:

Let A be the event that the job will be completed on time and B be the event that it rains.
Then we have,
P (B) = 0.45,
P (no rain) = P (B′) = 1 – P (B) = 1 − 0.45 = 0.55

By multiplication law of probability,
P (A | B) = 0.42
P (A | B′) = 0.90

Since, events B and B′ are subsets of the sample space S, therefore, by total probability theorem,
P (A) = P (B) P (A | B) + P (B′)
P (A | B′) = 0.45 × 0.42 + 0.55 × 0.9
= 0.189 + 0.495 = 0.684