1) The first one is a warm-up problem. I got it from Wikipedia (but it's no longer there):
Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of Bowl #1?This is a thinly disguised urn problem. It is simple enough to solve without Bayes's Theorem, but good for practice.
2) This one is also an urn problem, but a little trickier.
The blue M&M was introduced in 1995. Before then, the color mix in a bag of plain M&Ms was (30% Brown, 20% Yellow, 20% Red, 10% Green, 10% Orange, 10% Tan). Afterward it was (24% Blue , 20% Green, 16% Orange, 14% Yellow, 13% Red, 13% Brown).
A friend of mine has two bags of M&Ms, and he tells me that one is from 1994 and one from 1996. He won't tell me which is which, but he gives me one M&M from each bag. One is yellow and one is green. What is the probability that the yellow M&M came from the 1994 bag?3) This one is from one of my favorite books, David MacKay's "Information Theory, Inference, and Learning Algorithms":
Elvis Presley had a twin brother who died at birth. What is the probability that Elvis was an identical twin?To answer this one, you need some background information: According to the Wikipedia article on twins: ``Twins are estimated to be approximately 1.9% of the world population, with monozygotic twins making up 0.2% of the total---and 8% of all twins.''
4) Also from MacKay's book:
Two people have left traces of their own blood at the scene of a crime. A suspect, Oliver, is tested and found to have type O blood. The blood groups of the two traces are found to be of type O (a common type in the local population, having frequency 60%) and of type AB (a rare type, with frequency 1%). Do these data (the blood types found at the scene) give evidence in favour [sic] of the proposition that Oliver was one of the two people whose blood was found at the scene?5) I like this problem because it doesn't provide all of the information. You have to figure out what information is needed and go find it.
According to the CDC, ``Compared to nonsmokers, men who smoke are about 23 times more likely to develop lung cancer and women who smoke are about 13 times more likely.''6) And finally, a mandatory Monty Hall Problem. First, here's the general description of the scenario, from Wikipedia:
If you learn that a woman has been diagnosed with lung cancer, and you know nothing else about her, what is the probability that she is a smoker?
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say Door A [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say Door B, which has a goat. He then says to you, "Do you want to pick Door C?" Is it to your advantage to switch your choice?The answer depends on the behavior of the host if the car is behind Door A. In this case the host can open either B or C. Suppose he chooses B with probability p and C otherwise. What is the probability that the car is behind Door A (as a function of p)?
If you like this problem, you might also like the Blinky Monty Problem.
Solutions next week!
Please let me know if you have suggestions for more problems. An ideal problem should meet at least some of these criteria:
1) It should be based on a context that is realistic or at least interesting, not too contrived.
2) It should make good use of Bayes's Theorem -- that is, it should be easier to solve with BT than without.
3) It should involve some real data, which the solver might have to find.
4) It might involve a trick, but should not be artificially hard.
If you send me something that is not under copyright, or is usable under fair use, I will include it in the next edition of Think Stats and add you to the contributors list.