Question 1.
$15$ coupons are numbered $1,2,\ldots,15$ respectively. $7$ coupons are selected at random one at a time with replacement. What is the probability that the largest number appearing on a selected coupon is $9$?
My answer: $\left(\frac35\right)^7$
My textbook says this is the correct answer. My thought behind this was there are $9$ favorable outcomes for each draw. So probability for $3$ draws $=\left(\frac9{15}\right)^3$
Question 2:
A box contains tickets numbered $1$ to $20$. $3$ tickets are drawn from the box with replacement. The probability that the largest number on the tickets is $7$
My answer: $\left(\frac7{20}\right)^3$
My textbook gives the answer as $\left(\frac7{20}\right)^3-\left(\frac6{20}\right)^3$
It follows the method given in the video below.
https://youtu.be/lZMM6KzVRVQ
(The audio is in hindi and everyone might not be able to understand it, but you can see how the guy is solving the problem.)
My question is why the method that worked for question 1 not working for question 2.
Best Answer
For the first one: Let $P_n$ be the probability that the largest draw is $≤n$. Of course, the probability that any given draw is $≤n$ is $\frac n{15}$, thus $$P_n=\left( \frac n{15} \right)^7$$
Of course, the probability that the max is at exactly $n$ is then $P_n-P_{n-1}$, so in this case the answer you want is $$\left( \frac 9{15} \right)^7-\left( \frac 8{15} \right)^7$$
Which is not the answer you report. Your answer would be correct if you are just requiring that the maximum draw be $≤9$ but that isn't the question. Perhaps you mistyped the first question?
The second problem can be answered along the same lines.