# Bayes Theorem Formula

## Bayes Theorem Formula

An effective formula for estimating conditional probabilities is Bayes Theorem. Probabilities in the posterior are computed using it. Bayes theorem estimates the likelihood of an event based on circumstances that might be relevant to the event.

The Bayes Theorem Formula, for example, can be used to determine the likelihood that a diagnosis is correct given an observation that a patient has a specific symptom.To put it another way, let’s say a doctor is curious about a patient’s age and wants to know if they have cancer. The Bayes Theorem Formula can be used to acquire a more precise likelihood that a patient has cancer if age and malignancy are related.

## What is Bayes Theorem?

Simply put, the Bayes Theorem Formula calculates the conditional probability of event A, given that event B has already happened. It is applied in conditional probability calculations. Based on the hypothesis, the Bayes theorem determines the probability. Let’s now present the theorem and its justification.

According to the Bayes Theorem Formula, the conditional probability of event A given the occurrence of an additional event B is equal to the sum of the likelihood of B given A and the probability of A.

The formulas for total and conditional probabilities will be used to demonstrate the Bayes Theorem. When there is a lack of information regarding event A, the total probability of the event is estimated using the probabilities of occurrences related to event A. The likelihood of event A, given the occurrence of other related events, is known as conditional probability.

### Solved Example

Calculate P(B/A) if P(A/B) = 0.25, P(A) = 0.4 and P(B) = 0.5 using Bayes theorem.

Solution:

Given,

P(A/B) = 0.25

P(A) = 0.4

P(B) = 0.5

Using Bayes Theorem Formula

P(B|A) =

P(A|B)P(B)P(A)

P(B|A) =[0.25 ×0.50]/0.4

= 0.3125