# Bayes' Theorem Calculator

Updated P(A|B): 0.0000

# Bayes' Theorem

Bayes' theorem is a fundamental concept in probability theory and statistics. It allows us to update the probability of a hypothesis based on new evidence.

## The Formula

Bayes' theorem is expressed as:

P(B|A) * P(A)

P(B)

- P(A|B) - Posterior probability of hypothesis A given evidence B.
- P(B|A) - Probability of evidence B given hypothesis A.
- P(A) - Prior probability of hypothesis A.
- P(B) - Probability of evidence B.

## Example

Let's consider a medical test:

- P(A) = Probability of having a disease = 0.01 (1%)
- P(B|A) = Probability of testing positive given having the disease = 0.95 (95%)
- P(B|¬A) = Probability of testing positive given not having the disease = 0.02 (2%)

Now, let's calculate the probability of actually having the disease given a positive test result:

P(¬A) = 1 - P(A) = 0.99 (99%)

P(B) = P(B|A) * P(A) + P(B|¬A) * P(¬A) = 0.95 * 0.01 + 0.02 * 0.99 = 0.0297 (2.97%)

P(B|A) * P(A)

P(B)

Even with a positive test result, the probability of having the disease is around 32.1% due to the low prevalence of the disease.

## Conclusion

Bayes' theorem helps us make more informed decisions by incorporating new evidence into our prior beliefs. It is widely used in various fields including medicine, finance, and machine learning.