Last week’s puzzle was the toenail cancer test. The correct answer is that your chance of having toenail cancer is just under 1.9% obtained as follows. Of 20,000 people, 20 have toenail cancer (1 in 1000). Of those 20, 19 test positive (the test is 95% effective) but of the remaining 19,980 there will be 999 that test positive (the test is only 95% effective here too, meaning 5% of people without the disease test positive). So 999+19 = 1018 people test positive, and of them only 19 have toenail cancer. So your chance of having toenail cancer is 19/1018 which is 1.866%. This is much lower than intuitively people think (especially if they’ve just been told they’ve tested positive) and also much lower than physicians think since Bayes theorem is not a part of medical training (but should be). Test for rare events like this, even if seemingly effective, are overwhelmed by false positives (if toenail cancer only occured in 1 in 100,000 people, then the chance you have toenail cancer given you tested positive is approximately 1 in 500).
In any given year the weather station in New York’s central park records a certain rainfall. Ignore any trends and just assume that rainfall in one year is independent of rainfall the preceding year. A record year is one in which the rainfall exceeds that of any preceding year. Measurements started in 1835. In what year would you expect to get to the 20th record year (and as a hint, over the 160 year period from 1835 to 1994 there were 6 records).
Answer next week