The idea of “total probability” is a fundamental principle in statistics and probability theory used to determine the likelihood of an occurrence by considering all conceivable ways or scenarios in which the event may occur. It is frequently employed when you have knowledge of the likelihood of an event occurring under various settings or scenarios. The law of total probability, a specific application of this notion, is closely connected to total probability.
According to the law of total probability, if the sample space is divided into mutually exclusive and exhaustive events (i.e., events that cover all possible outcomes and do not overlap), the probability of any event can be expressed as a weighted sum of the conditional probabilities of that event given the various scenarios.
Mathematically, for an event A and a partition of the sample space into events B1, B2, B3,… Bn
the law of total probability is expressed as:
Where:
P(A) is the probability of event (A).
P(A|Bi) is the conditional probability of an event P( given that the event P( has occurred.
is the probability of an event P( (part of the partition).
The basic concept is to analyze all alternative scenarios (represented by the many events in the partition), compute the conditional probabilities for the event of interest within each scenario, and then weight these conditional probabilities by the likelihood of each scenario occurring. The overall probability of the occurrence P(A) is obtained by adding these weighted conditional probabilities.
Bayes rule:
Total probability is connected to Bayes’ Rule and is frequently used in combination with it to update probabilities in the presence of uncertainty or new information. The denominator in Bayes’ Rule is computed using total probability, which is critical for updating the conditional probability of a hypothesis depending on new information.
The rule of total probability is employed in this situation to account for all alternative scenarios or hypotheses P(Hi), allowing you to compute the total probability of the evidence P(E) by summing over all these possibilities. This is significant because P(E) is the normalization factor in Bayes’ Rule, guaranteeing that the posterior probabilities are appropriately scaled.