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Math Help - Consecutive letters

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    Consecutive letters

    There is a sequence of ten letters, consisting of two letters A and T, with equal probabilities for both.

    What is the probability that there will be atleast two As consecutively?
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    Quote Originally Posted by nahduma View Post
    There is a sequence of ten letters, consisting of two letters A and T, with equal probabilities for both.

    What is the probability that there will be atleast two As consecutively?
    Hi Nahduma,

    It's probably easier to find the probability that the sequence does not contain two consecutive A's, then subtract from 1.

    There are 2^{10} sequences of 10 letters each of which is an A or a T, and all the sequences are equally likely.

    Let's see if we can count the number of sequences which do not contain two consecutive A's. More generally, let's see if we can count the number of sequences of length n which do not contain two consecutive A's; call this number a_n. For brevity, let's say a sequence is "acceptable" if it does not contain two consecutive As.

    It's easy to see that
    a_1 = 2 and
    a_2 =3.

    Now suppose we have an acceptable sequence of length n+1 where n > 2. Break the possible sequences into two classes depending on whether the last letter is an A or a T. If the last letter is a T, then the preceding n letters can be any acceptable sequence; there are a_n of these. If the last letter is an A, then the nth letter must be an T, but the first n-1 letters can be any acceptable sequence; so there are a_{n-1} of these. Consequently,

    a_{n+1} = a_{n-1} + a_n

    Using this equation and what we already know about a_1 and a_2, you should be able to compute a_{10}. Then the probability that a sequence of length 10 does not contain two consecutive As is

    a_{10} / 2^{10},

    and the probability that it contains at least two consecutive As is

    1 - \frac{a_{10}}{2^{10}}.

    P.S.: Here's something for you to think about: What has all this got to do with the Fibonacci numbers?
    Last edited by awkward; February 9th 2010 at 05:19 AM. Reason: Added PS
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