Bayes’ Theorem is a fundamental concept in probability theory that describes the probability of an event based on prior knowledge of conditions that might be related to the event.

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It is named after the Reverend Thomas Bayes, who introduced the theorem.

The formula for Bayes’ Theorem is expressed as:

[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} ]

Here’s the breakdown of the terms:

- ( P(A|B) ): The probability of event A occurring given that event B has occurred (posterior probability).
- ( P(B|A) ): The probability of event B occurring given that event A has occurred (likelihood).
- ( P(A) ): The prior probability of event A.
- ( P(B) ): The prior probability of event B.

Bayes’ Theorem is particularly useful in updating probabilities based on new evidence. It is often applied in Bayesian statistics, machine learning, and various fields where probabilistic reasoning is involved.

**Example:**

Consider a medical example where ( A ) is the event of having a particular disease, and ( B ) is the event of testing positive for the disease.

- ( P(A|B) ): Probability of having the disease given a positive test result.
- ( P(B|A) ): Probability of testing positive given that the person has the disease.
- ( P(A) ): Prior probability of having the disease.
- ( P(B) ): Prior probability of testing positive.

The formula allows us to update our belief about the probability of having the disease after considering the test result.

Bayes’ Theorem is a powerful tool for updating probabilities as new evidence becomes available. It has applications in various fields, including medical diagnosis, spam filtering, and machine learning algorithms such as Bayesian networks.