In statistics and probability, the Bayes theorem plays a vital role in determining the probability of an event that is based on the other event that already has occurred. Reverend Thomas Bayes proposed the Bayes Theorem. Through this article, we will learn some basic concepts of the bayes theorem such as its meaning, statement, formula, and major terminologies used in the Bayes theorem.
What is meant by Bayes Theorem?
Bayes Theorem is the famous theorem used in probability to determine the conditional probability of an event A given that event B has already occurred. This specific theorem is also known as Bayes Law or Bayes Rule. Bayes Theorem is a theorem which is also known as the formula for the probability of “causes” because of its usage to determine the probability of an event based on the occurrences of the prior events. It is used to determine conditional probability.
What is the Bayes Theorem Statement?
The statement of Bayes Theorem is stated as:
Consider E1, E2,…, En as a set of events which are associated with the sample space S. Here all the events have no probability of occurrence and also these events form a partition of S. Now consider A to be an event to be associated with S, the according to Bayes Theorem,
P(Ei│A) = P(Ei)P(A│Ei)k=1n1P(Ek)P(A|Ek)
where, I = 1, 2, 3, 4,….,n
What is the Formula for Bayes Theorem?
The formula of the Bayes theorem generally exists for the events and random variables. This specific formula used in the Bayes theorem is derived from the definition of conditional probability. The Bayes theorem formula can be derived for events A and B, and also for continuous random variables X and Y.
Below-given is the formula used in the Bayes theorem for events A and B:
P(A|B) = P(A∩B)/P(B)
Here, P(A|B) is denoted as the probability of condition when event A is still occurring whereas event B has already occurred.
P(A∩B) is the indication of the probability of both, event A and Event B.
P(B) is the symbol denoting the probability of event B.
Terminologies Related to Bayes Theorem:
Now that we know the basics of Bayes theorem, let’s learn some major terms related to the concept of Bayes theorem:
- Conditional Probability: It is a type of probability where the probability of an event A is based on the occurrence of event B and is denoted by P(A|B).
- Joint Probability: It is a type of probability that measures the probability of two or more than two events occurring together and at the same time it is denoted by P(A∩B).
- Random Variables: Random variable also known as the experimental probability is considered as the real-valued variable the possible values of which are determined by any random experiment.
- Hypotheses: Hypotheses in probability are known as the events like E1, E2,…, En.
- Priori Probability: Priori probability of hypothesis Ei is considered as the probability P(Ei). It is the probability of an event that is computed before all the event-related information has been accounted for.
- Posteriori Probability: Posteriori probability of hypothesis Ei is considered as the probability P(Ei|A). It is the probability of an event that is computed after all the event-related information has been accounted for. Posteriori probability can also be called conditional probability.
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