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shw2 - Solutions to Homework 2 1 a Let us consider a...

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Solutions to Homework # 2 1. a) Let us consider a sequence of n 4 outcomes of tosses of a coin. The sequence may start with T , in which case there are no three heads in a row if and only if there are no three heads in a row in the subsequence of the last n - 1 outcomes, or with HT , in which case there are no three heads in a row if and only if there are no three heads in a row in the subsequence of the last n - 2 outcomes, or with HHT , in which case there are no three heads in a row if and only if there are no three heads in a row in the subsequence of the last n - 3 outcomes, or with HHH , in which case we get three heads in a row. Hence we get the formula a n = a n - 1 + a n - 2 + a n - 3 . We observe that a 1 = 2, a 2 = 4, a 3 = 7. Answer: we have a n = a n - 1 + a n - 2 + a n - 3 , where a 1 = 2, a 2 = 4 and a 3 = 7. b) Using the formula of Part a), we compute a 4 = 13, a 5 = 24, a 6 = 44, a 7 = 81, a 8 = 149, a 9 = 274, and a 10 = 504. Hence the probability that there will be three or more

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shw2 - Solutions to Homework 2 1 a Let us consider a...

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