This time from the Newsletter of the European Mathematical Society. The reviewer is Paulo Ventura Araujo, a mathematician at the University of Porto in Portugal. Interestingly, he notes at the start of the review that he had never heard of the Monty Hall problem prior to reading my book. Here are the final two paragraphs of the lengthy review:
The non-mathematical portions of the book, and even the least demanding mathematical portions, are very good reading and are suitable for a large non-specialist audience. In the cognitive chapter, for instance, we are made aware of two basic types of faulty probabilistic reasoning: that which is innate in the human mind and that which arises from misapplication of imperfectly learned mathematical tools. The conclusion, perhaps, is that a smattering of badly digested learning is often worse than no learning at all. The philosophical chapter delves deeply into a disturbing question: could it be that the best strategy in the long run, as shown by computing the relevant probabilities, is not the most advisable to adopt if you are only in for a single round of the game? Thankfully the answer is no, but before reaching this conclusion the author does a masterful job of dissecting a philosopher's arguments to the contrary.
And, since the author is a mathematician (but also a very skilled writer), it is somehow a vindication of our profession, after the embarrassing events that surrounded the Monty Hall problem, the he came to write this fine book.
Score! I especially appreciated the first sentence of the review, since that was precisely what I was going for.
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"The philosophical chapter delves deeply into a disturbing question: could it be that the best strategy in the long run, as shown by computing the relevant probabilities, is not the most advisable to adopt if you are only in for a single round of the game?"
How could it NOT be? Suppose I ask you to pay $100,000 for a 1 in 1 million chance of $1 trillion. You'd be an IDIOT to accept that offer; you are virtually guaranteed to give me $100,000 for nothing.
But if I made the offer to a million people, I'd lose money.
No amount of subtle argumentation can get around this obvious fact: Sometimes a positive expected utility can be a very bad idea.
What embarrassing events surrounded the Monty Hall Problem?
Valhar - I think Araujo is referring to the response to Vos Savant, which included numerous mathematicians telling her she was wrong.
Patrick, given the unexpressed assumption that each of the million people is given a unique number between 1 and 1 million, there will be a guaranteed winner of your 1$ trillion. In your example, you've only collected 0.1$ trillion, so yes you lose.
No surprise; when charities use this method, the numbers are smaller; with about 2 orders of magnitude reduction of the pay-out ratio.
The more interesting case would be where the odds are 1 in 1 million for each of the million people. Do the odds of no-one's winning balance the odds of more than one winner? Consider that it is improbable, but possible that every ticket will win.
Slightly OT, but related: Anyone's got a take on the doomsday argument? Something goes seriously wrong in there, but I don't know enough probability theory to explain what:
http://en.wikipedia.org/wiki/Doomsday_argument
Not as specialist, but: the Doomsday argument is flawed since everyone is told to assume they are typical, so the results are contradictory: at each time of application you get different answers. But maybe if you averaged all the estimates of the total from everyone doing it (or at least, averaged the estimates made at each time in sequence), then they'd average out to the correct value.
BTW, here's another probability trick: if you have unlimited amounts to bet, you can mop up by betting amounts in a sequence like 1, 3, 9, 27, ... no matter how small the odds of winning. That's because the series always sums to less than what you win (assuming at least double return.) Hence you spend 1 + 3 + 9 = 13 and win back 18, and so on. It's risky with fixed amounts since in the long run, you'll lose enough complete runs not to have any net gains. But it's still IMHO a paradox since you're cheating the average expectation value.