# oracle aide

## January 3, 2013

### Using Venn Pie Chart to illustrate the classic Bayesian mammography problem.

Filed under: html, statistics — Tags: — oracleaide @ 8:42 pm

Here is a link to the promised Venn Pie Chart “Applet” and its screenshot:

Some comments for the updated HTML5 Venn Pie Chart  at dropbox.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 grey 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 grey 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).

TBD: Show the applet inside the post.