Bayes' theorem calculator

The Basics in Plain Language

Bayes’ Theorem helps you adjust your beliefs based on new information. Think of it like a math tool for answering: “How likely is my guess now that I’ve seen the evidence?”

Imagine you’re trying to figure out if it’ll rain today. Bayes’ Theorem uses three key pieces of information:

1. Your initial guess (e.g., 20% chance of rain).
2. How likely the evidence is if your guess is true (e.g., 90% chance of dark clouds when it rains).
3. How often the evidence happens in general (e.g., 10% chance of dark clouds on any day).

The formula combines these to give you an updated probability:

Try the Calculator

This tool lets you solve for any missing value. Just fill in three percentages (0–100%) and select what to calculate:

Example:
If you see dark clouds (evidence), the calculator might tell you the rain chance jumps from 20% to 64%.

Real-Life Examples

1. Medical Tests: Why “95% Accurate” Can Mislead

• Prior: Only 1% of people have Disease X.
• Likelihood: Test is 95% accurate for sick patients.
• False Alarms: Test is 5% wrong for healthy people.
• Total Evidence:
  $(95\% \times 1\%) + (5\% \times 99\%) = 5.9\%$
• Updated Belief:
  $\frac{95\% \times 1\%}{5.9\%} \approx 16\%$
  A positive test means just 16% risk, not 95%!

2. Spam Emails: How “Free” Triggers Filters

• Prior: 2% of emails are spam.
• Likelihood: 80% of spam emails say “free.”
• False Alarms: 0.1% of real emails say “free.”
• Updated Belief:
  $\frac{80\% \times 2\%}{(80\% \times 2\%) + (0.1\% \times 98\%)} \approx 94\%$
  An email with “free” has a 94% spam chance.

Step-by-Step Calculator Guide

Scenario: You want to know the chance of having a rare allergy (1% prior) after testing positive (test is 90% accurate for true cases, 8% false positives).

1. Input Prior: 1% (how common the allergy is).
2. Input Likelihood: 90% (test accuracy if you’re allergic).
3. Input Total Evidence:
   $(90\% \times 1\%) + (8\% \times 99\%) = 8.82\%$
4. Calculate Posterior:
   $\frac{90\% \times 1\%}{8.82\%} \approx 10.2\%$
   Result: A positive test means only a 10% chance you actually have it!

Common Mistakes to Avoid

1. Ignoring the Base Rate: Don’t forget the starting probability (e.g., rare diseases stay rare even with positive tests).
2. Confusing “Accuracy”: A test’s “95% accuracy” doesn’t mean a 95% chance you’re sick—it depends on how common the disease is.
3. Forgetting False Positives: Always ask, “How often does this evidence happen by accident?”.

Why Bayes’ Theorem Matters Today

• AI & Netflix Recommendations: Updates predictions based on what you watch.
• Self-Driving Cars: Adjusts decisions using real-time sensor data.
• COVID Testing: Helps interpret results in low-risk vs. high-risk groups.

FAQ

Can I use percentages instead of decimals?

Yes! The calculator works with 0–100% inputs (no need for 0.05 = 5%).

What if I don’t know “Total Evidence”?

Select “Calculate P(E)” in the tool. It uses:
$P(E) = (P(E|H) \times P(H)) + (\text{False Positive Rate} \times (100\% - P(H)))$

Does Bayes’ Theorem work for multiple updates?

Absolutely! Use the posterior (updated belief) as your new prior for the next piece of evidence.