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Practical Examples

See how Bayes' Theorem reveals surprising truths and clarifies complex questions

Counter-Intuitive Scenarios

These examples demonstrate how our intuitions about probability can be dramatically wrong. Test your gut feeling, then see what the mathematics reveals.

1

The Medical Test Paradox

A rare disease affects 0.1% of the population (1 in 1,000 people). There's a highly accurate test:

  • 99% sensitivity: detects the disease 99% of the time when present
  • 99% specificity: correctly identifies healthy people 99% of the time

You test positive. What's the probability you actually have the disease?

2

The Monty Hall Problem

You're on a game show with three doors. Behind one is a car, behind the others are goats. You pick Door 1.

The host (who knows what's behind each door) opens Door 3, revealing a goat. He then asks:

"Do you want to switch to Door 2, or stick with Door 1?"

What gives you the better chance of winning the car?

3

The Taxi Cab Problem

A city has two taxi companies:

  • • Green Cabs: 85% of all taxis
  • • Blue Cabs: 15% of all taxis

A taxi was involved in a hit-and-run at night. A witness identified it as Blue. The court tested the witness's ability to identify taxi colors at night and found they were correct 80% of the time.

What's the probability the taxi was actually Blue?

%

Advanced Applications

Beyond counter-intuitive puzzles, Bayes' Theorem helps us understand complex real-world questions where different starting assumptions lead to different conclusions—even when everyone agrees on the evidence.

4

The Resurrection Question: How Priors Shape Conclusions

This example demonstrates one of the most important insights from Bayesian reasoning: rational people can reach different conclusions from the same evidence when they start with different prior beliefs.

Consider the historical question: Did Jesus of Nazareth rise from the dead? Let's examine how three different perspectives—each using valid Bayesian reasoning—reach dramatically different conclusions.

Note: This example is purely educational and demonstrates mathematical reasoning. We present multiple perspectives respectfully without advocating for any particular position.