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polymathwannabe comments on Open thread, July 21-27, 2014 - Less Wrong Discussion
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Question about Bayesian updates.
Say Jane goes to get a cancer screening. 5% prior of having cancer, the machine has a success rate of 80% and a false positive rate of 9%. Jane gets a positive on the test and so she now has a ~30% chance of having cancer.
Jane goes to get a second opinion across the country. A second cancer screening (same success/false positive rates) says she doesn't have cancer. What is her probability for having cancer now?
According to your percentages, out of every 10,000 women, 5% = 500 have cancer and 95% = 9,500 do not.
Of those 500 women with cancer, 80% = 400 will get a positive test and 20% = 100 will get a negative one.
Out of those 9,500 women without cancer, 9% = 855 will get a positive test and 91% = 8,645 will get a negative one.
After taking the first test, Jane belongs to the group of 1,255 women out of every 10,000 who have a positive test.
Of those 1,255 women, 400 have cancer. Jane's likelihood of having cancer is 400/1,255 = 31.87%.
If we take those 1,255 women to a second test, 80% = 320 of the 400 women with cancer will get a positive test and 20% = 80 will get a negative test.
Of those same 1,255 women with a first positive test, 9% = 77 of the 855 women without cancer will get a positive test and 91% = 778 will get a negative test.
After taking the second test, Jane belongs to the group of 858 women out of every 10,000 with one positive and one negative test.
Of those 858 women, 80 have cancer. Now Jane's likelihood of having cancer is 80/858 = 9.32%.