What is the difference between conditional probability and Bayes Theorem ?

Conditional probability and Bayes' theorem are related concepts in probability theory, but they have distinct meanings and applications.

Conditional Probability:
Conditional probability refers to the probability of an event occurring given that another event has already occurred. It quantifies the likelihood of an event happening based on some additional information or condition. It is denoted as P(A|B), which reads as "the probability of event A given event B."

For example, let's say we have two events A and B. The conditional probability of event A given event B is calculated as:

P(A|B) = P(A ∩ B) / P(B)

This formula states that the probability of both events A and B occurring (P(A ∩ B)) divided by the probability of event B happening (P(B)) gives the conditional probability of event A given event B.

Bayes' Theorem:
Bayes' theorem is a fundamental result in probability theory that provides a way to update the probability of an event based on new evidence or information. It allows us to revise our initial beliefs or probabilities in light of new data. Bayes' theorem is named after the Reverend Thomas Bayes, who formulated it.

The formula for Bayes' theorem is:

P(A|B) = (P(B|A) * P(A)) / P(B)

In this formula:

P(A|B) is the conditional probability of event A given event B (what we want to calculate).
P(B|A) is the conditional probability of event B given event A.
P(A) is the probability of event A.
P(B) is the probability of event B.
Bayes' theorem states that the probability of event A given event B is equal to the conditional probability of event B given event A, multiplied by the prior probability of event A, divided by the prior probability of event B.

Bayes' theorem is particularly useful when dealing with inverse probabilities or when we want to update our beliefs based on new information. It is commonly applied in fields such as statistics, machine learning, and data analysis.

In summary, conditional probability is the probability of an event given another event has occurred, while Bayes' theorem is a mathematical formula that allows us to update probabilities based on new evidence or information. Bayes' theorem incorporates conditional probability as one of its components.