My commenters, thank them, quite nicely outlined the major reasons that someone in the Deal or No Deal problem I posited would be wiser to take the Banker’s offer of a sure $11,750 rather than to keep a randomly selected one of $1, $10, $7,500, $25,000, or $35,000. Even though the expectation value, the average that the Contestant could expect from sticking with her suitcase if she played the game an enormous number of times is $13,502.20, fairly noticeably larger than the Banker’s offer, she is just playing the game the once. She’s more likely to do worse than the Banker’s offer, and is as likely to do much worse — $1 or $10 — rather than do any better.
If we suppose the contestant’s objective is to get as much money as possible from playing, her strategy is different if she plays just the once versus if she plays unlimitedly many times. I don’t know a name for this class of problems; maybe we can dub it the “lottery paradox”. It’s not rare for a lottery jackpot to rise high enough that the expected value of one’s winnings are more than the ticket price, which is typically when I’ll bother to buy one (well, two), but I know it’s effectively certain that all I’ll get from the purchase is one (well, two) dollars poorer.
It also strikes me that I have the article subjects for this and the previous entry reversed. Too bad.
Of course, when I put the problem to my students on the test, I had the “Why?” and braced myself for the surprises which come up when mathematics students are asked to write prose. There are subjects where essays are expected, and mathematics is not among them, and “why” questions aren’t often put to the students. Most put forth that the Contestant should reject the Banker’s offer, on the grounds of the expectation value being higher than the offer. A substantial minority put forth that the Contestant should take the sure thing, on the grounds she was more likely to do worse than better. Nearly all were taken aback by the idea that I was accepting both “take the offer” and “don’t take the offer” as responses, as long as the justification for it was given.
There’s probably a lesson there that I should have taken closer to heart in setting up problems after that. It’s challenging enough to create a word problem in which the student is asked to calculate something that a person might reasonably wish to know, represented by some problem which might actually happen. But an important part of mathematics is the setting up of a model: deciding what one wants to know, and what pieces of information about the whole world are relevant enough to be included. That’s also quite hard to test, especially when the course description just says you should come out knowing how to calculate weighted means and to do a test of statistical significance.
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