From Maura:
Started probability today. I went through the basics: what probability (chance) is; some simple examples (coin tosses, drawing from a deck of cards, rolling a die) and some fuzzy examples (predicting whether or not the Red Sox will win the World Series, predicting the weather for tomorrow). We stuck to the “simple examples” and eventually saw the pattern: the probability of an event happening is the number of ways the event could happen divided by the total number of events. We can (and often do) turn this into a percentage. Along the way we see that we can’t have a probability that is more than 100% or less than 0% – just not possible.
So far so good. Then we talked about a simple raffle example: 500 tickets, one prize worth $1000. Someone noticed right away that the tickets need to sell for at least $2 each. We worked out fair price and got a bit mixed up on average value and expected value (this happens to me every time); then we worked it out using the probability of winning $998 and the probability of losing $2. This is the way to do it and we tried it again with a $3 raffle ticket. It was good to do the calculation as just about everyone got it wrong (so what I mean is that it was good to revisit the orders of operations and make sure we knew how to do this on a calculator or by using Google).
Then we looked at some Massachusetts lottery statistics. If there is a 1 in 751 million chance of winning the jackpot, what is the probability of winning? We figured out that it was awfully small – around the order of 10^(-7). Then I gave them another situation to consider: suppose I found a T pass on the subway and (let’s assume someone from the Boston area lost it) decided to systematically dial every number in the 617 area code. What is the probability that on any particular call, I reach the person who lost the ticket? We took some time to figure out that there are 10^7 possible phone numbers in the 617 area code, then figures out the probability that one of the calls is to the right person – this is on the order of 10^(-5). Then we talked about it and made the comparison: it’s more likely that one of these (random) phone calls will reach the right person than we’ll win that jackpot for Mega Millions. I must confess that this is not my idea. I got it from a nice book I picked up called “Five Minute Mathematics”. But it was fun to turn it into an example in class!
The last bit we did was the Massachusetts Lottery exercise from the book. They really struggled with figuring out lottery revenues given that 23% of total lottery revenues is $761 million. A reminder that percentages are really really tough.
From Ethan:
Started probability today. Decided to reorder the book presentation, for the sake of variety and to try to wake up the class, which has been pretty drowsy and unresponsive lately. At least today most showed up.
I started with coin flips, 50:50, probability 1/2. Easy stuff that no one complained about. When I asked how often you’d get two heads with two coins one student guessed 10% of the time, another 25%. The second guess is right, the first not, but both were just guesses. With a little struggle we managed to list the four possibilities (HH, HT, TH, TT) and believe that therefore 1/4 was right. (It helps to imagine two coins – a nickel and a dime – rather than one coin flipped twice. The independence is much more intuitive.) It took quite a while to get the class to list all eight possibilities for three coins, but they managed. I asked about 10 heads in a row. They reluctantly accepted that the probability would be 1/2^10, but didn’t remember (from just last week) the approximation 2^10 ~ 1000.
In order to make the point (which I will have to make repeatedly) that rare things do happen – just rarely – I asked them to imagine what would happen if we asked all 15,000 UMass students to flip a coin ten times and report the results. About one in every thousand, or 15, would see 10 heads in a row. Those particular students would think they were somehow special, but looking at the big picture there’s nothing odd about what happens.
Then I did the grid experiment. I passed out sheets of paper with an empty 7×10 grid of squares and asked them to fill in the squares with H or T by imagining flipping a coin. Although I said (clearly, I thought) that they should go left to right in each row and top to bottom row by row, I noted that several of them filled the squares in some other order – or in a “random” order. When all were done I asked about counting runs of four (all H or all T). Since we were doing science, they agreed that we should predict the answer before we counted. There are 67 possible starting places for a run of four (since when one row ends you go on to the next), and HTTTTTH would count as two runs of four T. About 1/16 of the 67 possible runs should be HHHH and about 1/16 TTTT, so we expect about 1/8 of 67 ~ 8 runs of four, on average. When we compiled data from the class it looked something like this:
number of runs number of students 0 1 1 2 2 1 3 4 4 3 ... 10 1
The average number of runs (a weighted average of the table entries) is much less than 8. That’s because people are psychologically conditioned to make the tosses “even up” – four heads in a row seems too unlikely.
When I asked them to make the same count column by column rather than row by row the number of runs of four was much larger – still averaging to less than 8, though.
We had about 20 minutes left to talk about lotteries. They understood why you might be willing to pay $1 for one of 100 tickets when the prize was just, say $35, if the rest of the money went to the PTA running the raffle. They also understood that you could enjoy gambling at a casino even if you were bound to lose (on average in the long run) because you enjoyed the thrill of thinking you might win.
Started to look at the problem in the book on the state lottery. The class got stuck immediately trying to compute total lottery revenue when $761 million (23%) was given to cities and towns as local aid.
Sigh.
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