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MIT6_042JF10_rec20_sol

MIT6_042JF10_rec20_sol - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science November 24, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 20 Philosophy of Probability Applying probability to real-world processes often involves a little bit of philosophy. Let’s first consider this simple problem: What is the probability that N = 2 6972607 1 is a prime number? One might guess 1 / 10 or 1 / 100. Or one might get sophisticated and point out that the Prime Number Theorem implies that only about 1 in 5 million numbers in this range are prime. Or one can say that assigning a probability to this statement is nonsense because there is no randomness involved; the number is either prime or it isn’t. This question highlights the distinction between two philosophical approaches to prob- ability. One school of thought says that probabilities can only be meaningfully applied to repeatable processes like rolling dice or flipping coins. In this view, the probability of an event represent the fraction of trials in which that event will occur. This view is sometimes called classical statistics, sampling theory, or the frequentist approach. An alternate view is the Bayesian approach, in which a probability can be interpreted as a degree of belief in a proposition. A Bayesian would agree that the number above is either prime or composite, however would be perfectly willing to assign a probability to each possibility. The Bayesian approach is thus broader and willing to assign probabilities to any event, repeatable or not. One challenge with the Bayesian approach is coming up with reasonable prior probabilities for events that only occur once. As an aside, it is not clear whether Bayes himself was Bayesian in this sense. However, a Bayesian would be willing to talk about the probability that Bayes was Bayesian while a sampling theorist would say that is nonsense because there is no repeatable process that generates Bayes’ beliefs! Getting back to prime numbers, there is a probabilistic primality test due to Rabin and Miller. If N is composite, there is at least a 3 / 4 chance that the test will discover this. (In the remaining 1 / 4 of the time, the test is inconclusive; it never produces a wrong answer.) Moreover, the test can be run again and again and the results are independent. So if N actually is composite, then the probability that k = 100 repetitions of the Rabin-Miller do not discover this is at most: 100 1 4
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2 Recitation 20 So 100 consecutive inconclusive answers would be
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