### Exploring Bayes' theorem with Venn Pie Chart

Probabilities, %:

P(A):

P(B|A):

P(B|A'):

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This is an applet for a blog post

### Using Venn Pie Charts to illustrate Bayes' theorem

at oracleaide.wordpress.com.

Here is a classical Bayesian problem from the famous Yudkowsky's "An Intuitive Explanation of Bayes' Theorem":

• 1% of women at age forty who participate in routine screening have breast cancer.
• 80% of women with breast cancer will get positive mammographies.
• 9.6% of women without breast cancer will also get positive mammographies.
• A woman in this age group had a positive mammography in a routine screening.
• What is the probability that she actually has breast cancer?

• The Venn Pie Chart (I admit - I made up the term) describes events presented in the problem using colored overlapping sectors:
• The pink sector represents frequency of the first event (A), women having breast cancer.
• The gray sector represents frequency of the second event (B), women having a positive test.
• The area where sectors overlap (the dark gray sector), represents frequency of the second event given that the first one has happened (B|A or "B given A").
• Ratios of sector areas (or arcs) represent probabilities of events.
• How is it different from Venn diagrams or Pie Charts?
• It is different from a regular Pie Chart because its sectors overlap.
• It is different from a regular Venn (or, rather, Euler) diagram because it presents sets using sectors, not circles.
• Just click the "Draw" button, and observe that:
• Percentage of women with breast cancer - the pink sector - is still just 1%.
• The most confusing number in the whole problem is 80%.
It is the dark gray sector - representing women with positive test and cancer. And it covers only part (80% to be precise) of the tiny pink sector, not the whole circle (a.k.a. universe).
• There is a lot of false positives - the light gray sector, covering 9.6% of the white area (not the whole circle!).

• Feel free to explore different combinations of probabilities: (50,50,50), (25,25,25), (25,50,25), (30,30,30), (50,0,50).