Unlocking Bayes' Theorem: Intuitive Examples of Lies and Disease Diagnosis

1 Bayes Theorem

Bayes' theorem, also known as the "inverse probability theorem", allows us to compute the probability of an event when we know related conditional probabilities.

We first look at conditional probability, then rearrange terms to obtain Bayes' formula.

When an event B is partitioned into k parts, P(A) can be expressed in the law of total probability.

Because Bayes' theorem can feel counter‑intuitive, we will use concrete examples to deepen understanding.

2 Example 1 – Is the person lying?

Assume a person tells the truth two out of three times. He rolls a die and reports that the result is 4. What is the probability that the die actually shows 4?

B: the statement is true

A: the reported result is 4

The calculation yields a probability of less than 30% that the die truly shows 4.

3 Example 2 – Disease Diagnosis

In a population, 1 in 10,000 people has a certain disease. A test is highly accurate: if a person is healthy, it yields a false‑positive result with probability 2%; if a person is diseased, it yields a false‑negative result with probability 1%. If a randomly selected person tests positive, what is the probability that they actually have the disease?

A: test result is positive

B: person has the disease

Applying Bayes' formula shows that, despite a positive test, the probability of actually having the disease is very small because the disease is rare (the prior probability is 0.01%). The numerator is tiny, while the denominator is large due to many false positives.

4 Summary

This article illustrated the application of Bayes' theorem through two simple examples. By carefully considering prior probabilities, one can correctly interpret seemingly paradoxical results, and the theorem can be applied to many real‑world situations.