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

1. Aug 24, 2014

### mahler1

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
2. Aug 24, 2014

### Ray Vickson

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
3. Aug 24, 2014

### mahler1

It is exactly as you've said, sorry for not making myself clear.

4. Aug 24, 2014

### Orodruin

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