The document explains Bayes' theorem by using a hypothetical example of a test for nose cancer. It states that 1 out of 100 people have nose cancer and the test is 98% accurate. For someone who tests positive, it calculates through Bayes' theorem that the likelihood they have nose cancer is approximately 33%, compared to their original 1% risk before taking the test. This Bayesian update demonstrates how additional information received from a test result can change the probability assessment.

1. Understanding
Bayes’Theorem
By David Siegel

2. 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:

3. You test positive.
What is the likelihood
that you have the disease?
Here is the problem:
Please work out your answer before continuing …

4. People
who have
the
disease:
1%
A priori:

5. True
positives:
1% * 98%
False
positives:
2%
False negatives
Test accuracy: 98%
1% * 2%
After testing everyone:

6. True
positives:
980
False negatives:
Total population:
100,000
20
False
positives:
2,000
It helps to use numbers:

7. Chance you
have nose
cancer
True positives
=
All positives
Given that you tested positive:

8. Chance you
have nose
cancer
True positives
=
All positives
This is Bayes’Theorem!

9. Chance you
have nose
cancer
980
=
980 + 2,000
Plug in the numbers:

10. Chance you
have nose
cancer
980
=
2,980
= 32.88%
Do the math:

11. 33%!
Chances that you have nose
cancer, given that you tested
positive:

12. Before test
1%
After test
33%
This is called a Bayesian update:
update

13. What if your test had
been negative?
What is the chance you
have nose cancer now?
Extra-credit question:

14. Understanding
Bayes’Theorem
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