Sequence of LettersDate: 04/23/99 at 05:06:27 From: Robles Mellin Subject: Theory of runs problem Hi. I have tried without success to solve the following problem: Suppose you have an alphabet of 10 letters and a sequence of 600 letters, what is the probability of finding the subsequence DDDD ? For instance: AJFKDSKDDDDYIYITIOTOPDLKJDSLKJDSSDSDFLKFDLKFD ____ There are: 10^600 possible sequences. How can I find the probability that subsequence DDDD appears in those 10^600 sequences? Thank you. U. Robles Date: 04/23/99 at 14:36:03 From: Doctor Anthony Subject: Re: Theory of runs problem Without going through some very involved combinatorics we could approach the problem in a purely probabilistic way. Imagine drawing letters one at a time, each letter having an equal probability of 1/10 at each draw. We require first the expected number of draws to the first occurrence of 4 consecutive D's. Let a = expected number of draws to first D. We must make 1 draw at least and we have probability 9/10 of returning to a, so a = 1 + 9/10 a (1/10)a = 1 a = 10 Let E = expected number of draws to 2 consecutive D's. Consider that we have just drawn a D and what happens on the next draw. We are dealing with the (a+1)th draw. With probability 9/10 this is not a D and we return to E. So E = (a+1).1 + (9/10)E note we must have at least a+1 draws so p = 1 (1/10)E = a+1 E = 10(a+1) and now putting in the value a=10 we get E = 10(11) = 110 The expected number of letters to 2 consecutive D's is 110. The equation for 3 consecutive D's is: E = (110+1).1 + (9/10)E (1/10)E = 111 and E = 1110 The equation for 4 consecutive D's is: E = (1111).1 + (9/10)E (1/10)E = 1111 and E = 11110 So the expected number of letters to 4 consecutive D's is 11110. In 600 letters the expected number of occurrences of 4 D's is 600/11110 = 0.0540054 Using the Poisson distribution with parameter m = 0.0540054 we get P(0) = e ^(-0.0540054) = 0.947427 and probability of at least one occurrence of 4 D's is 1 - 0.947427 = 0.052573 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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