l24_spcl_topics - 6.042/18.062J Mathematics for Computer...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science May 12, 2005 Srini Devadas and Eric Lehman Lecture Notes Special Topics 1 Streaks Was the table of H s and T s below generated by flipping a fair coin 100 times, or by someone tapping the H and T keys in a what felt like a random way? HTTTHTHTTHTTHTHTHTHT TTTHHTHHTHTHTTHHHTHT HHTHHTTTHHHTHTTHHHHT THTTHHTHTHTHTHTTHTHH HTTHHHTHTHHHTHTHHHTH There is no way to be sure. However, this sequence has a distinctive feature that is com- mon in random human-generated sequences and unusual in truly random sequences: namely, there is no long streak of Hs or Ts. In fact, no symbol appears above more than four times in a row. How likely is that? If we flip a fair coin 100 times, what is the probability that we never get five heads in a row? 1.1 From a Probability Problem to a Counting Problem The sample space for this experiment is { H,T } 100 ; that is, the set of all length-100 se- quences of Hs and Ts. If the coin tosses are fair and independent, then all 2 100 such se- quences are equally likely. Therefore, we need only count the number of sequences with no streak of five heads; given that, the probability that a random length-100 sequence contains no such streak is: Pr ( sequence has no HHHHH ) = # sequences with no HHHHH 2 100 This is a common situation. We have reduced a probability problem to a counting problem. Unfortunatley, we have no hope of solving the counting problem by direct computation. No computer can consider all 2 100 sequences of H s and T s, keeping track of how many lack a streak of five heads. But, on the bright side, there is a big bag of mathematical tricks for solving counting problems. In this case, well use a recurrence equation . The recurrence equation approach involves two steps: 2 Special Topics 1. Solve some small problems. 2. Solve the n-th problem using preceding solutions. Lets see how this approach plays out in the analysis of streaks. 1.2 Step 1: Solve Small Instances Let S n be the set of length- n sequences of H s and T s that do not contain a streak of five heads. Our eventual goal is to compute | S 100 | . But for now, lets just compute | S n | for some very small values of n : | S 1 | = 2 ( H and T ) | S 2 | = 4 ( HH , HT , TH , and TT ) | S 3 | = 8 | S 4 | = 16 | S 5 | = 31 ( HHHHH is excluded!) These are called base cases ....
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l24_spcl_topics - 6.042/18.062J Mathematics for Computer...

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