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 $=k^{n}$
$\Rightarrow P\left(x \leq k\right)=\left(\frac{k}{N}\right)^{n}$
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 $=P\left(x \leq k\right)-P\left(x\right)=\frac{k^{n} - \left(k - 1\right)^{n}}{N^{n}}$