# Math Help - Drawing balls randomly with replacement

1. ## Drawing balls randomly with replacement

An urn contains $n$ balls numbered $1$ through $n$. If you withdraw $m$ balls randomly in sequence, each time replacing the ball selected previously, find $P[X=k], k=1,2,...,m$, where $X$ is the maximum of the $m$ chosen numbers

I'm not even sure what this question is asking me to find. Can someone help me clarify the question please?

2. ## Re: Drawing balls randomly with replacement

Originally Posted by I-Think
An urn contains $n$ balls numbered $1$ through $n$. If you withdraw $m$ balls randomly in sequence, each time replacing the ball selected previously, find $P[X=k], k=1,2,...,m$, where $X$ is the maximum of the $m$ chosen numbers

I'm not even sure what this question is asking me to find. Can someone help me clarify the question please?
What is the probability that the maximum of m balls drawn with replacement is k?

CB

3. ## Re: Drawing balls randomly with replacement

Originally Posted by I-Think
An urn contains $n$ balls numbered $1$ through $n$. If you withdraw $m$ balls randomly in sequence, each time replacing the ball selected previously, find $P[X=k], k=1,2,...,m$, where $X$ is the maximum of the $m$ chosen numbers
I will discuss one case. You can generalize.
Suppose $\{1,2,3,4,5,6\}$ is the set of numbered balls and $m=4.$

Let’s find $P(X=3)$. There $3^4$ ways to have 4-tuples containing only 1, 2, or 3.
There $2^4$ ways to have 4-tuples containing only 1 or 2.
Therefore there $3^4-2^4$ ways to have 4-tuples made of 1, 2, or 3 with at least one 3.

Then $P(X=3)=\frac{3^4-2^4}{6^4}$