## HCOL 196, January 28, 2011

We first discussed the remaining Fermi problems. On the parking lot problem, most teams just based their answer on personal experience with parking lots; one team actually tried to estimate the time for a space to come free from the fact that you could command 20 spaces. I commented that since one might arrive randomly in that interval, maybe dividing by two would be reasonable (though the other parameters are uncertain enough that this is probably not necessary). On the Grand Canyon, the most convincing calculation indicated that what I had heard is false. The population problem was fairly straightforward, and most teams correctly thought that the number of millionaire households would be a few percent (the actual number is 5.5%). Small numbers were guessed for the piano tuners, I recall that it is something like ten in the phone book (but they serve the entire county). And the value of the rice is much, much bigger than the U.S. National Debt.

We then looked at a problem where you pick one of three coins out of a hat; one has two heads, one has two tails, and one is a regular coin. If the coin is tossed, and comes up heads, what is the probability that the other side of the coin is heads (that is, that you picked the two-headed coin)? First we identified the states of nature, which are best described as which coin we have. Before tossing the coin, we have no data, so the prior on each is 1/3. (The likelihood will rule out that we have the two-tailed coin). The likelihood of seeing heads if the coin is HH is 1, and of seeing heads if it is HT is 1/2. Thus the tree that pops out is as shown below. It is twice as likely that the coin has two heads than that it has a head and a tail.

This problem is equivalent to the “Bertrand’s box” paradox. The problem can also be done as a “spreadsheet” calculation: Finally we looked at the “Monty Hall” problem, which is described here on WikiPedia. Another, completely equivalent version is the “Three Prisoners” problem. The important thing here is that Monty knows where the prize is, and always opens a door with a goat, and always offers the contestant a chance to switch. Under those circumstances, it is an advantage to switch, as our probability tree shows: 