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Even with grade inflation, there are thirty marks of A or A- or B+ or B to look at, each one attached to a different course like Advanced Calculus or 20th Century Philosophy or Introduction to Economics.
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(Peter Winkler, “Coin Flipping,” Communications of the ACM 56:11, 120.)Īn observation by Oxford University mathematician Nick Trefethen:Ī student leaves university in America with a transcript full of information. There are 10 such sequences one is HHHTT. “Solving for X yields a startlingly high 62 flips.” To get your expected dues down to $32, you need to pick a sequence where this “head start” effect doesn’t obtain. “If X is the average time needed to get HHHHH starting fresh, the average of 1 + X and 1 is 32,” Winkler writes.
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But if you choose HHHHH (for example), one occurrence of this outcome gives a huge head start to the next - if the next flip is a tail, then you’re starting over cleanly, but if it’s a head then you’ve already produced the next occurrence. “Overlapping causes problems.” It is true that in an infinite sequence of random flips, the average distance between one occurrence and the next of any fixed sequence is 1/32. What sequence should you pick?Īt first it seems that it shouldn’t matter - any fixed sequence should have the probability (1/2) 5, or 1/32. dollars for instance, if you choose HHHHH and it takes 36 flips to produce a run of five heads, then your annual dues will be $36. Your dues will be the total number of flips in U.S. You’ll name a head-tail sequence of length 5, and then a coin will be flipped until that sequence appears in five consecutive flips. Walker, “Arrogance,” American Philosophical Quarterly 35:4, 379-390.)Ī curious puzzle by Dartmouth mathematician Peter Winkler: You’ve just joined the Coin Flippers of America, and fittingly the amount of your dues will be decided by chance. … So if Darcy and Kissinger believe that they are doing pretty well by the standards of human excellence, it is not obvious that they are wrong, and their arrogance must therefore consist in something other than a false belief.” “Henry Kissinger, for instance, is by all accounts a highly arrogant person, but his intellectual talents are considerable, and all philosophical accounts of the good life for human beings assign such talents an important role. University of Minnesota philosopher Valerie Tiberius asks, what then is arrogance? He is superior to the people he’s judging, even according to their own standards. The trouble is, he’s right - Darcy is educated, intelligent, wealthy, and handsome.
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Could you expect me to rejoice in the inferiority of your connections? To congratulate myself on the hope of relations whose condition in life is so decidedly beneath my own?” Of Elizabeth Bennet he says, “She is tolerable but not handsome enough to tempt me and I am in no humour at present to give consequence to young ladies who are slighted by other men.” Later, when she spurns his proposal, he tells her directly, “Nor am I ashamed of the feelings I related. Darcy arrogant, proud, and conceited because of his reactions to village people. In Jane Austen’s Pride and Prejudice, the residents of Longbourn consider Mr.