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Math Help - Probability: Arrangements Question

  1. #1
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    Probability: Arrangements Question

    Question: 5 people designated as A, B, C, D, E are arranged in linear order. Assuming that each possible order is equally likely, what is the probability that


    a) there is exactly one person between A and B

    b) there is exactly 2 people between A and B

    c) there is exactly 3 people between A and B

    I am not sure how to work out this problem. I am having trouble coming out with the possible ways of outcomes and I do not know what to do with them. Here is what I have so far:


    a) there is exactly one person between A and B

    There is 3 ways that will happen ACBDE, ADBCE, AEBCD
    but will ACBED count as another way?

    Then there is 120 ways that these 5 people can line up.

    Would we have to divide 3 by 120 to get the proability?


    Thank you very much for your help.
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  2. #2
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    Quote Originally Posted by hotmail590
    Question: 5 people designated as A, B, C, D, E are arranged in linear order. Assuming that each possible order is equally likely, what is the probability that


    a) there is exactly one person between A and B

    b) there is exactly 2 people between A and B

    c) there is exactly 3 people between A and B
    (a)The possibilities are,
    AXBXX This is (1)(3)(1)(2)(1)=6
    BXAXX This is (1)(3)(1)(2)(1)=6
    XAXBX ................................
    XBXAX
    XXAXB
    XXBXA
    Total=6 (times total arragement =6)=36

    The number of ways to arrange 5 people is
    (5)(4)(3)(2)(1)=120
    Thus, probability is 36/120=3/10
    ---------
    (b)
    The possibilities are,
    AXXBX This is (1)(3)(2)(1)(1)=6
    BXXAX This is (1)(3)(2)(1)(1)=6
    XAXXB ....................................
    XBXXA
    Total=6(times total arragement=4)=24
    The number of ways to arrange 5 people is
    (5)(4)(3)(2)(1)=120
    Thus, the probability is 24/120=1/5=2/10
    ---------
    (c)This only happens when,
    AXXXB =6
    BXXXA =6
    In total we have 12, thus the probability is,
    12/120=1/10

    Thus the answers respectively are,
    30% , 20% , 10%
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