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Apr
3
2010
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The total number of possible orderings for the N women is N!, and, of these, the number of orderings with the very best candidate in the jth position is (N-1)! for any given j, to account for all possible permutations of the other N-1 candidates . Of these orderings, only a fraction would result in the best candidate actually being selected (using the strategy of passing on the first k candidates and then selecting the next “best so far”). Given that the best candidate is in position j, the necessary and sufficient condition for her to be selected is for the best candidate in the first j-1 positions to be in the first k positions. Hence the fraction of the (N-1)! orderings with the best candidate in the jth position that would result in the best candidate being selected is k/(j-1). Dividing the result by N!, we see that the probability of selecting the optimum woman at the jth round, for j in the range k+1 to N, based on some particular value of k, equals the product
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Thus, as k increases, the left side approaches 1, and we can take the exponential of both sides to give
Jan
11
2008
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But on the whole, it's been a great thing. Pick up any conference proceedings from the last 20 years, in the fields of math, computer science, physics, or chemistry (among numerous others), and you'll see the results of TeX layout. Pick up a book published by Springer-Verlag, and it's almost certainly typeset by TeX. Look at Greg Chaitin's books - every one was written using TeX. Look at any typeset equation in pretty much any published source, from websites to conference proceedings, to journals, to textbooks. If the equation looks really good, if everything is in exactly the right place, and every symbol is correctly drawn in relation to everything else - odds are, it was generated by TeX. Even hardcore Microsoft word users generally use something TeX based for doing equations.
Jan
7
2008
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In fact the most successful systems that humans have been able to design are ones in which most of the design effort goes into letting the system be able to grow in a fashion that detects and corrects the error, so that the system doesn't come apart. The Internet is a good example of that, because it's something that didn't require any central control and wasn't constructed, the way a clock is constructed. Another good example is the American Constitution, because the people who designed it realized that it would be very difficult to write laws for how people should live 50 years from their time and place, so they wisely made most of the Constitution a way of dealing with error situations that would come up and a way of keeping bad things from propagating. They didn't try to tell people how they should live over the next two centuries.
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The reason our group has been successful is that our whole development system is designed to allow us to late-bind things that we discover along the way, things we would not have to find out along the way if we had a real engineering discipline.
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