Class, 10/1

We discussed the fourth problem set, which was done pretty well by you all. I pointed out several errors that were made:

One group forgot, on the second problem, that three different widgets were sampled independently, so that the likelihood had three factors in it, not one.

One group didn't recognize that the third problem had only two states of nature, that is, whether it is Urn #1 or Urn #2. Somehow this group ended up with five states of nature, that is, 1R, 1W, 2R, 2W and 2B, where the number is the urn number and the letter the color. The point is that the thing you don't know and want to learn is always the way to determine what the states of nature are. Here, what we don't know is which urn we've picked, so that tells us what the states of nature are.

One group got the states of nature right, but in the third ball selection forgot that it is the number of balls in the urn when the ball is picked that gives the denominator. True, this is a step made without replacement, but since the first two steps all involved returning the ball (that is, with replacement), there are still ten balls in the urn when the third ball is picked.

On the last problem, one group computed correctly the contribution to the likelihood for each word, but then added them instead of multiplying to get the likelihood. Since the likelihood is the probability that we got 3 of the first word AND 5 of the second AND 3 of the third, you have to multiply. When you compute the probability of one thing AND the probability of another thing, you always multiply. Addition is for when you want to compute the probability of one thing OR the probability of another thing. For example, when you add the joint probabilities in a spreadsheet calculation, you are computing the probability of (data,SON1) OR the probability of (data,SON2) OR ..., to get the probability of the data, irregardless of which SON is true.

We then finished the "Trewel" problem and calculated that the best thing for Alan to do is to shoot in the air, letting Bob and Charlie duke it out, and then, with one of them dead for sure, to come in on his second try and try to kill the survivor. We also recalculated the probability of Alan eventually killing Bob when Alan goes first. See Class, 9/29 for a calculation.

Finally, we discussed the first item on the study sheet, Fermi problems. We discussed the length of the Nile river...and I reminded everyone about the geometric mean trick. If you can put a reasonable lower bound on a quantity and a reasonable upper bound, so that you are pretty sure that the true value is between those bounds, then a decent guess at the correct value is to multiply the lower bound by the upper bound and take the square root of that number. For the Nile, a lower bound might be 100 miles and an upper bound 10,000 miles, which would give an estimate of 1000 miles. Wikipedia says 4100 miles, so this is not a great estimate. For the Mississippi, we imagined the U.S. as a box 3000 miles wide and 2000 miles high, so a length of 2000 miles. The actual length is 2340 miles, so that worked better.

I pointed out that the important thing with regard to Fermi problems is how you got the answer, not the actual value of the answer.