Understanding Bayes’ Theorem

By David Siegel

1 out of 100 people has nose cancer, a fictional disease

1

A new test is 98% accurate.2

You test positive. 3

What is the likelihood that you have the disease?

4

Nose cancer!

Here is the problem:

You test positive.

What is the likelihood that you have the disease?

Here is the problem:

Please work out your answer before continuing …

People who have the disease: 1%

A priori:

True positives: 1% * 98%

False positives: 2%

False negatives

Test accuracy: 98%

1% * 2%

After testing everyone:

True positives: 980

False negatives:

Total population: 100,000

20

False positives: 2,000

It helps to use numbers:

Chance you have nose cancer

True positives=

All positives

Given that you tested positive:

Chance you have nose cancer

True positives=

All positives

This is Bayes’ Theorem!

Chance you have nose cancer

980=

980 + 2,000

Plug in the numbers:

Chance you have nose cancer

980=

2,980= 32.88%

Do the math:

33%!

Chances that you have nose cancer, given that you tested positive:

Before test1%

After test33%

This is called a Bayesian update:

update

What if your test had been negative?

What is the chance you have nose cancer now?

Extra-credit question:

Understanding Bayes’ Theorem

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