I'm confused, and surely wrong, about the cancer example.
1 in 10000 people are sick. 1 sick person : 9999 well persons multiply by 100: 100 sick people : 999900 well persons 99% of the sick people have positive tests: (0.99 * 100 = ) 99 Positive tests 1% of the well people have false positive tests: (0.01 * 999900 = 9999)
Using the odds view: number of sick persons with positive tests / total number of persons with positive tests: (99 / (99 + 9999) = 99 / 10098. Multiply top and bottom by (1/99) => (99/99) / (10098/99) = 1 / 102. The text says the answer is 1 / 101.010101… which is 99/10000.
So, try the waterfall method.
prior odds of being sick: 1 in 10000. Being sick: 1 Being well: 9999
chance of having positive test while sick: 99 chance of having positive test while well: 1
odds of being sick given positive test: (1 / 9999) * (99 / 1) = 99 / 9999 = 0.00990099 probability of being sick given positive test: 99 / (99+9999) = 1 / 102 from above.
Where did I go wrong? Thanks in advance for any time you have!