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Math Help - random variables

  1. #1
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    random variables

    A store has found that 10% of the items it sells are retruend after Christmas. On one day, a total of 50 items are sold by 5 clerks: Clerk A sold 5 items, Clerk B sold 20 items, and the rest were sold by the other 3 clerks.

    a) what is the probability that none of the items sold by Clerk A will be returned?
    ans: 0.5905

    b) what is the probability that exactly 3 of the items sold by Clerk B will be returned?
    ans: 0.190

    c) what is the probability that between 2 and 7 (inclusively) of the items sold by the other 3 clerks will be returned?
    ans: 0.727

    d) what is the probability that between 3 and 6 (inclusively) of the items sold by Clerk B and that exactly 4 of the items sold by the other 3 clerks will be returned?
    ans: 0.044
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  2. #2
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    Quote Originally Posted by skhan
    A store has found that 10% of the items it sells are retruend after Christmas. On one day, a total of 50 items are sold by 5 clerks: Clerk A sold 5 items, Clerk B sold 20 items, and the rest were sold by the other 3 clerks.

    a) what is the probability that none of the items sold by Clerk A will be returned?
    ans: 0.5905
    This used binomial probability.
    {n \choose m}p^m(1-p)^{n-m}
    Over here you need all 5 not to. The probability of that not happening is 90% thus,
    {5 \choose 5}(.9)^5(.1)^0=.5905

    Quote Originally Posted by skhan
    b) what is the probability that exactly 3 of the items sold by Clerk B will be returned?
    ans: 0.190
    {20 \choose 3}(.1)^3(.9)^{17}\approx .1901

    Quote Originally Posted by skhan
    c) what is the probability that between 2 and 7 (inclusively) of the items sold by the other 3 clerks will be returned?
    ans: 0.727
    Here you have 25 of them, thus you need to find,
    \sum ^7_ {k=2} {25 \choose k }(.1)^k(.9)^{25-k}\approx .7265
    Quote Originally Posted by skhan
    d) what is the probability that between 3 and 6 (inclusively) of the items sold by Clerk B and that exactly 4 of the items sold by the other 3 clerks will be returned?
    ans: 0.044
    This is probability of two events. First find between 3 and 6 which is,
    \sum^6_{k=3}{20\choose k}(.1)^k(.9)^{20-k}\approx .3207
    And then probability exactly 4 which is,
    {25 \choose 4}(.1)^4(.9)^{25-4}\approx .1384
    Now multiply them to get,
    \approx .04438.


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    CaptainBlack for the nice trick for combinations using LaTeX really helped.
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  3. #3
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    wow thanks!
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