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Probability problem (counting numbers which are not divisible by ##k##

  1. Aug 24, 2014 #1
    1. The problem statement, all variables and given/known data

    Suppose one extracts a ball from a box containing ##n## numbered balls from ##1## to ##n##. For each ##1 \leq k \leq n##, we define ##A_k=\{\text{the number of the chosen ball is divisible by k}\}.##

    Find ##P(A_k)## for each natural number which divides ##n##.

    3. The attempt at a solution

    I thought of thinking of ##P(A_k)## as ##P(A_k)=1-P({A_k}^c)##. And ##P({A_k}^c)=\{\text{the number of the chosen ball is not divisible by k}\}##. If ##k\geq 3##, then I know how many numbers less than ##k## are coprime with ##k## (Euler's totient function), however, in this case I would need the numbers greater than ##k## which are not divisible by ##k##. I don't know how to count them. I couldn't think of anything else, any advice or suggestions would be appreciated.
     
    Last edited: Aug 25, 2014
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  3. Aug 24, 2014 #2

    Ray Vickson

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    If I understand what you are saying, ##P(A_2)## is the probability of choosing an even number, while ##P(A_3)## is the probability of choosing a multiple of 3, ##P(A_4)## is the probability of choosing a multiple of 4, etc. Is that right? As it stands, there seems to be an ambiguity: ##A_2 \cap A_4 \neq \emptyset##, etc. Would that be OK?
     
    Last edited: Aug 24, 2014
  4. Aug 24, 2014 #3
    It is exactly as you've said, sorry for not making myself clear.
     
  5. Aug 24, 2014 #4

    Orodruin

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    A number is divisible by k if it is a multiple of it ...
    What is the largest m such that ##mk \leq n##?
     
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