From Maura
I reviewed a bit of the homework, including the Mass Lottery problem that we had started at the end of class last time. My memory from having assigned this before is that students tend to make it more difficult than it really is.
After that we started Chapter 12, looking at the probability of independent events and repeated events. First we did the coin toss and talked through the probability of getting a lot of heads in a row – they eventually saw the pattern. Some students immediately drew a tree diagram, which is a good way to organize the possible outcomes. From there we did the grid experiment, basically following the approach that Ethan used. Except I just had them do it – we didn’t do any prediction or even talk about what we may be looking for. Even thought I told them to imagine flipping a coin each time, some students told me afterward that they kept alternating heads and tails. The distribution came out as in Ethan’s class: very few runs across (in fact, they were really surprised) and a good number when we looked down columns.
From there I talked a bit about the “double or nothing” game and we worked that out, then I finished by looking again at independent events. We used the geometric approach of looking at probabilities as areas. Then the probability of two independent events both happening is the area of the resulting rectangle. This is a very nice approach: not only is it geometric, but it gives students a clear visual image of what the probability of something not happening looks like.
While I think these chapters are very important, I think they still need work in terms of making the ideas more relevant. We’ll see how it goes with Chapter 13.
From Ethan
I spent nearly an hour on the Mass Lottery homework problem. One or two students managed to draw the proper pie chart (part (d)) even though they felt (were) stuck on all the rest. In fact the pie chart is a reasonable place to begin the problem, so I was able to discuss “read the whole thing through and make sense of what you can” as a better strategy than “I can’t do part (a) so I give up.”
I did finally work through “23% of what is $761 million”, using the pie chart as a prop. Still painful for too many of them, but at this point that’s their responsibility. I have done as much as I can.
Estimating average per person lottery spending was an eye opener – if 2/3 of the 6 million people in the commonwealth play then each buys about $800 worth of tickets in a year! We discussed the lottery as a regressive tax, since poor people play disproportionally. Is it immoral of me not to play, knowing that?
Aware of Maura’s warning, I computed the fair price of a PTA lottery ticket (1000 $1 tickets for one $300 prize means each is worth 30 cents on average), then showed that you needed to know only the total collected and the total given out in prices to know that the fair price for a ticket was 30% of its face value. Then we easily saw that the fair value of a $5 Mass Lottery ticket was 69% of $5, or about $3.50.
I need to update the answer to part (e).
With the less than half an hour left we talked about “house advantage” in more general terms – first roulette, returning $2 for a successful bet on red. That was clearly fair for a 36 segment wheel half red half black. Zero and 00 change the winning probability to 18/38 < 1/2 from 18/36 = 1/2, but with the same $2 payout. That’s the house advantage. All understood it. I didn’t make it quantitative. I did point out that it was much less than the house advantage in the lottery.
We closed with a quick analysis of (hypothetical) collision insurance. Suppose it costs $600 (per year) with a $500 deductible. Then there’s a 2×2 table that describes your cost:
buy insurance don't buy $3000 accident $500+$600 $3000 no $3000 accident $600 $0
One of the students (and not the strongest of them) realized that what you should do depended on the probability of an accident. Again, I didn’t quantify, which I think was the right choice. I did say that I bought collision insurance only for the first year owning a new car (and that now I bought only used cars), and that I was “ahead” after 50 years of driving. I need to rewrite the insurance section of the probability chapter to include computations like these.
In insurance calculations the house advantage pays the insurance company’s overhead and profit – like the casino they see the whole population, not the individual gambler. I noted that Harvard was big enough to be self-insured for fire (at least I think that was true once).
Next time: pari-mutuel odds, hundred year storms, why doubling up isn’t a winning strategy.
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