Question

A box contains tickets numbered 1 to N. $n$ tickets are drawn at random with replacement. Find the probability that the largest number on the selected ticket is $K$, $(K\le N)$

• A. $(\frac{K}{N}{)}^n$
• B. $(\frac{K}{N}{)}^{n-1}$
• C. $\frac{K^n-K^{n-1}}{N^n}$
• D. $\frac{K^n-(K-1{)}^n}{N^n}$

Answer 1

Answer: (D)

$\frac{K^n-(K-1{)}^n}{N^n}$

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 = $K^n$
However, $K$ may or may not have been drawn.
No. of ways of choosing numbers from $1$ to $(K-1)$ = ${(K-1)}^n$
Thus, no. of ways in which the maximum no. drawn is $K$ = $K^n-{(K-1)}^n$
Therefore, probability = $\frac{K^n-{(K-1)}^n}{N^n}$