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Math Help - probability to create a binary bit pattern

  1. #1
    Junior Member
    Joined
    Feb 2010
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    probability to create a binary bit pattern

    Hello,

    I did not anderstand this exercise.

    May you PLEASE help me?

    If we have created a binary bit pattern from 100 bits.

    What is the probability to create another binary bit pattern from 100 bits which will have distance smaller than 10 from the already created one.

    The distance is based on XOR calculation like i.e:

    010101
    101010
    --------
    111111 - > distance 6

    or

    010101
    010100
    --------
    1 -> distance 1

    Looking forward for your answer!
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  2. #2
    MHF Contributor
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    If the distance is less than 10, then it must be something between 0 and 9.

    You can create all possible 100 bit patterns from one bit pattern A by choosing the bits from A that change.

    That means that you need to choose less than 10 bits that change. More precisely, you need to choose exactly n bits that change, where 0 \leq n \leq 9.

    The number of ways to do this is
    \sum_{n=0}^9 {100 \choose n}.

    The total number of 100 bit patterns is 2^{100}.

    Hence, the required probability is

    \frac{\sum_{n=0}^9 {100 \choose n}}{2^{100}},

    assuming that each bit pattern is chosen with equal probability.
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  3. #3
    Junior Member
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    Feb 2010
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    I woud appreciate if you can let me know where I can read more about this kind of problems and in which category they fall in probability. Moreover could you explain what is the probability to create the same bit pattern?

    Thank you...
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  4. #4
    MHF Contributor
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    This is a question of combinatorics.
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