Bayes' Theorem Calculator

Calculate conditional probability P(A|B) using Bayes' theorem with prior and likelihood probabilities

Calculate Conditional Probability

Choose how to calculate the evidence probability P(B)

The prior probability of event A occurring

The probability of B given that A has occurred

The total probability of event B occurring

Bayes' Theorem Results

26.67%
P(A|B)
Posterior Probability
45.00%
P(B)
Evidence (Given)

Formula Used

P(A|B) = [P(B|A) × P(A)] / P(B)
P(A|B) = [60% × 20%] / 45.00%
P(A|B) = 12.0000 / 0.4500 = 26.67%

Interpretation

Given that event B has occurred, there is a 26.67% probability that event A has also occurred.

The prior probability of A was 20%, but observing event B increases our confidence that A occurred to 26.67%.

Example: Weather Prediction

Weather Example Problem

Question: What's the probability of rain given cloudy morning?

A = Rain event

B = Cloudy morning event

P(A) = 20% (probability of rain on any given day)

P(B) = 45% (probability of cloudy morning)

P(B|A) = 60% (probability of cloudy morning given rain)

Solution

P(A|B) = [P(B|A) × P(A)] / P(B)

P(A|B) = [60% × 20%] / 45%

P(A|B) = 12% / 45%

P(A|B) = 26.67%

There's a 26.67% chance of rain given a cloudy morning.

Bayes' Theorem Components

P(A|B)

Posterior probability

What we want to find

P(A)

Prior probability

Initial belief about A

P(B|A)

Likelihood

Probability of evidence given A

P(B)

Evidence

Total probability of evidence

Common Applications

🏥

Medical Diagnosis

Test accuracy and disease probability

📧

Spam Filtering

Email classification systems

🎯

Machine Learning

Naive Bayes classifiers

⚖️

Legal Evidence

Evidence evaluation in courts

💰

Risk Assessment

Financial and insurance analysis

Bayes' Theorem Tips

Use when you know P(A) and P(B|A) but need P(A|B)

Extended mode calculates P(B) from components

Posterior combines prior belief with new evidence

All probabilities must be between 0% and 100%

Understanding Bayes' Theorem

What is Bayes' Theorem?

Bayes' theorem is a mathematical formula that describes how to update the probability of a hypothesis when given new evidence. It's named after Thomas Bayes, an 18th-century statistician and theologian.

When to Use Bayes' Theorem

  • When you know the prior probability P(A)
  • When you know the likelihood P(B|A)
  • When you need to find P(A|B)
  • For updating beliefs with new evidence

The Bayes' Formula

P(A|B) = [P(B|A) × P(A)] / P(B)

Extended Formula

P(A|B) = [P(B|A) × P(A)] / [P(A) × P(B|A) + P(¬A) × P(B|¬A)]

Where:

  • • P(A|B) = Posterior probability
  • • P(A) = Prior probability
  • • P(B|A) = Likelihood
  • • P(B) = Evidence
  • • P(¬A) = Probability of NOT A