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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