Today we discussed the homework.

First the lottery problem. There were several things here that not everyone thought of. One important thing is that there were 200,000,000 tickets and a chance of 1/80,000,000 that a particular ticket would win. This means that in a series of such lotteries, we can expect 2.5 tickets to win on average, so you’d have to share your prize with 2.5 people, making the prize worth about 112,000,000 to you, not 280,000,000.

A refinement of this is to figure out the probability P(n) that there will be n=0, 1, 2,… other winners, and thus figure out the amount that you’d win in each of these cases, setting up a probability tree with many nodes instead of just putting $112 M. If p is the probability that a single ticket wins (p=1/80 M), then (1-p) is the probability than the ticket loses, and the probability that all N=200,000,000 tickets lose is (1-p)^{N}=0.0821. The probability that a specific ticket wins and all the others lose is p*(1-p)^{(N-1)}=p*0.0821, since there is hardly any difference between (1-p)^{N} and (1-p)^{(N-1)}. But there are N tickets out there, so the probability that one of them wins and all the others lose is N*p*0.0821=0.2052 (I just realized I wrote the wrong number on the whiteboard). For two, the probability that a particular two win, one buying the ticket first and the other later, and all the others lose, is p^{2}*0.0821; but there are N of the first and (N-1) of the second, giving a factor of N*(N-1), which is essentially N^{2}, and there are two orders in which the tickets could have been bought, so this has to be multiplied by 1/2, giving a probability for two winners other than yourself of (N*p)^{2}*0.0821/2=0.2565. In general, for k other winners, the probability is (N*p)^{k}*0.0821/k!. The calculation of the probabilities of the various branches is outlined here:

The first few of these are in the picture of the completed tree (without calculations):

But there are two other flies in the ointment. First is taxes: You don’t get to keep all the money, you have to give Uncle Sam 39% and some to Jim Douglas (if you live in Vermont). Second is annuitization: The only way you can get the full jackpot is to have the lottery buy an annuity for you that will pay you the amount over 20 years in equal installments. But if you take the money immediately (probably the best choice), they will only give you the amount that they have to pay the insurance company for the annuity, which is about half of the jackpot. So you will get about 0.5*0.6=0.3 of the amounts in the figure. So, if you are the only winner, your net take after taxes would be, not $112 M, but only $33.6 M. That’s the figure that really should be entered as the gain, and when you do this (and similarly for the other numbers), the “buy the ticket” branch will actually have a loss.

What does the lottery do with the $140 M that it doesn’t have to pay out? It uses it to finance its beneficiary, education mostly. So net, the lottery is actually a tax that people are willing to pay.

Then we turned to the lawsuit problem. Most groups did a pretty good job; there were some calculational glitches but they were minor. One group added a third branch, “just continue the lawsuit,” which wasn’t among the choices, but the tree says that this isn’t the best choice (I’ve added this below). The only unusual item in this tree is the second box that comes if the other side makes a counter-counter offer of $3 B. Since this choice comes later on in the logic, it is to the right of the first decision box. The final tree is here:

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