For calculating posterior probabilities (conditional probabilities under statistical dependence), the following information is available:

(A) Conditional probabilities

(B) Original probability estimates (prior probabilities) of mutually exclusive and collectively exhaustive events

(C) Arbitrary event with probability ≠ 0 and for which conditional probabilities are also known

(D) Joint probabilities of prior probability and conditional probability

Given the information that the arbitrary event has occurred, arrange the above information in a sequence of their requirement as per Bayes’ Theorem.​

Correct option is D

To calculate posterior probabilities using Bayes’ Theorem, the steps and corresponding sequence of information required are:
1. (B) Original Probability Estimates (Prior Probabilities):
· Begin with the prior probabilities of mutually exclusive events. These probabilities represent the initial estimates of how likely each event is, without considering the occurrence of any additional event.
2. (A) Conditional Probabilities:
· Use the conditional probabilities of the arbitrary event (which has occurred) given each prior event. These probabilities indicate the likelihood of the arbitrary event occurring under each prior event.
3. (D) Joint Probabilities of Prior and Conditional Probabilities:
4. Calculate the joint probabilities by multiplying prior probabilities by their respective conditional probabilities: Joint Probability =

5. (C) Arbitrary Event (with Probability ≠ 0):
· Using the joint probabilities, calculate the posterior probability for each event given that the arbitrary event has occurred. This is done using Bayes’ Theorem:

Information Booster:
1. Bayes’ Theorem Formula:

· P(A|B): Posterior probability (probability of A given B).
· P(A): Prior probability of event A.
· P(B|A): Conditional probability of B given A.
· P(B): Marginal probability of event B.
2. Key Steps in Applying Bayes’ Theorem:
· Start with prior probabilities of events (P(A)).
· Multiply by conditional probabilities (P(B|A)).
· Normalize by dividing by the total probability of event B to ensure the probabilities sum to 1.
Additional Knowledge:
1. Applications of Bayes’ Theorem:
· Medical Diagnosis: Estimating the probability of a disease given test results.
· Spam Filtering: Calculating the likelihood of an email being spam based on keywords.
· Risk Assessment: Evaluating probabilities in finance, insurance, and other industries.
2. Why Normalization Is Necessary:
· The total probability of the arbitrary event must be calculated to ensure posterior probabilities are consistent and sum to 1. This is achieved by dividing the joint probability by the marginal probability.