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Math Help - 9999 Notes

  1. #1
    hgg
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    Question 9999 Notes

    Hello to all,

    New to the forum.
    I have a question you might be able to answer.

    You have a box with 9999 notes with a single four digit number written on them.
    You have two favourite numbers, e.g. Number 1 and number 7.
    You pick one note from the box.

    What is the probability that from the 4 digits of the note, 3 of them are always
    a combination of your favourite numbers?

    For example 1118, 1716, 7771, 1131, 7277, 1717, 7777, 1111 etc.

    Thank you!
    George.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: 9999 Notes

    Hello George. Given 2 out of 10 favorite digits, the probability that there are k favorite digits out of n on the slip of paper is:

     P(k) =  \begin {pmatrix} n \\ k \end{pmatrix}0.2^k 0.8^{n-k}

    where   \begin {pmatrix} n \\ k \end{pmatrix} is the combination of k out of n items, which is equal to  \frac {n!}{k!(n-k)!}

    So here you want the probability that either 3 or 4 of the 4 digits are among your two favorites:

     P(3) + P(4) = \begin {pmatrix} 4 \\ 3 \end{pmatrix}0.2^3 0.8^1 + \begin {pmatrix} 4  \\ 4 \end{pmatrix}0.2^4 0.8^0 = 0.0256+0.0016 = 0.0272
    Thanks from hgg and mash
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  3. #3
    hgg
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    Re: 9999 Notes

    Hello abaines,

    Thank you for your fast reply!
    I only need the probability that either 3 of the digits are among the favourites.
    In the example 7777, one of the 7s is just another digit that happens to be a 7.

    So the probability should be 0.0256, correct?
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  4. #4
    MHF Contributor ebaines's Avatar
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    Re: 9999 Notes

    Quote Originally Posted by hgg View Post
    So the probability should be 0.0256, correct?
    No - 0.0256 is the probability of getting precisely three of your favorites and one that is not your favorite - for example it captures the cases of 7772 , 7773, 7774, etc, but not 7771 nor 7777. So you have to add the probabilty that all four are favorites as well.
    Thanks from mash
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  5. #5
    hgg
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    Re: 9999 Notes

    Oh, I see. Thanks again!
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