Total Probability Theorem

The probability of an event occurring is determined by probability. There are a variety of situations in which we must forecast the outcome of a real-life event. We may be certain or doubtful about the result of an event. In these situations, we say the event has a chance of happening or not happening. In games, business (for developing probability-based estimates), and artificial intelligence, probability has a wide range of applications. The probability formula can be used to calculate the likelihood of an event by dividing the favourable number of possibilities by the total number of possible outcomes. The chance of an event occurring can range from 0 to 1 because the number of favourable outcomes can never exceed the entire number of outcomes. There can’t be a negative number of positive outcomes, either. The total probability theorem can be used to calculate the chances of an event occurring based on the divisions of the sample space.

The probability of the event A occurring from the distinct partitions is 

P(A) = P(E1)P(A/E1) + P(E2)P(A/E2) + ……P(En)P(A/En). 

for a sample space divided into n partitions {E1, E2, E3, ……En} such that {E1 U E2 U E3,…..U En} = S.

Total Probability Theorem

The total probability theorem is useful for computing the likelihood of an event occurring, which is the result of the probabilities of this event occurring from the various partitions of the sample space. We suppose that the sample space S is divided into multiple partitions {E1, E2, E3, ……En} such that {E1 U E2 U E3…U En} = S, and that the probability of an event A occurring is the sum of the likelihood of this event occurring from the various partitions of the sample space.

The following three easy examples can help you understand this better.

• The doctor travels to the patient for treatment using various modes of transportation, and his chance of arriving on time is equal to the total of his chances of arriving on time using various modes of transportation.

• A student will compete in an external competition on behalf of the institution. The likelihood of picking a student for the competition is equal to the sum of the probabilities of selecting a student from each of the school’s classes.

• The likelihood of finding a defective mango is the total of the probabilities of finding a defective mango in each of the mango boxes.

The cornerstone of Bayes’ Theorem is the total probability theorem, which aids in estimating the reverse probability of an event occurring from partitions of a sample space S. The probability of an event occurring from a given partition of the defined space can be calculated using the theorem of total probability, and the reverse probability from a given partition of the sample space can be calculated using Bayes’ Theorem, and this is completed using the theorem of total probability. 

Statement of the Bayes’ theorem:

The events E1, E2, E3,……En are a collection of exhaustive events of a sample space S, such that {E1, E2, E3,……En} are the partitions of a sample space S, and the occurrence of the event A from the sample space S is P(A) = P(E1)P(A/E1) + P(E2)P(A/E2) + ……P(En)P(A/En)

Proof:

Given that E1, E2, E3,…En is a set of exhaustive events in the sample space S,

{E1 U E2 U E3, …..U En} = S

E1, E2, E3,…En etc. are also mutually exclusive events. These are occurrences in which the occurrence of one event prevents the occurrence of another set of events, and no shared events exist.

Ei n Ej = ∅

Consider an event A that occurs in the sample space S.

A = A ⋂ S

A = A ⋂ {E1 U E2 U E3, …..U En}

A = {A ⋂ E1} U {A ⋂ E2} U {A ⋂ E3}, …..U {A⋂ En}

Let’s put probability to the test on both sides of the equation.

P(A) = P(A ⋂ E1) U P(A ⋂E2) U P(A ⋂E3), …..U P(A⋂ En)

P(A) = P(A ⋂ E1) + P(A⋂E2) + P(A⋂E3), …..U P(A⋂ En)

P(A) = P(E1)P(A/E1) + P(E2)P(A/E2) + ……P(En)P(A/En)

• Conclusion:

The entire probability rule (also known as the Law of Total Probability) divides probability computations into manageable chunks. When you don’t know enough about A’s probabilities to calculate it directly, it’s used to find the likelihood of an occurrence, A. Instead, you use the likelihood of a related occurrence, B, to compute the chance of A.