Question

A box contains tickets numbered $1$ to $N$. $n$ tickets are drawn from the box with replacement. The probability that the largest number on the tickets is $k$ is

• A. ${\left(\frac{k}{N}\right)}^n$
• B. ${\left(\frac{k-1}{N}\right)}^n$
• C. $0$
• D. none of these

Answer 1

The correct option is D none of these

Let X denote the largest number on the n tickets drawn. We have
$P(X\le k)={\left(\frac{k}{N}\right)}^n$ and $P(X\le k-1)={\left(\frac{k-1}{N}\right)}^n$
Thus $P(X=k)={\left(\frac{k}{N}\right)}^n-{\left(\frac{k-1}{N}\right)}^n$