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

Answer: (D)

Number of ways of selecting $n$ tickets from $N$ tickets numbered from $1$ to $N$ with replacement $=N^n$
Since the tickets are drawn with replacement, the same ticket may be repeated. Since the largest ticket is $k$ , the tickets chosen are from $1$ to $k$ .
Total no. of ways of doing it $=k^n$
$\Rightarrow P\left(x\le k\right)={\left(\frac{k}{N}\right)}^n$
However, $k$ may or may not have been drawn.
No. of ways of choosing numbers from $1$ to $\left(k-1\right)={\left(k-1\right)}^n$
$\Rightarrow P\left(x\right)={\left(\frac{k-1}{N}\right)}^n$
Thus, no of ways in which the maximum no. drawn is $k=k^n-{\left(k-1\right)}^n$
Therefore, probability $=P\left(x\le k\right)-P\left(x\right)=\frac{k^n-{\left(k-1\right)}^n}{N^n}$