Suppose that the probability that a new seed planted in a specific farm germinates is equal to the proportion of all planted seeds that germinated in that farm previously. Suppose that the first seed planted in the farm germinated, but the second seed planted did not germinate. For positive integers n and k (k < n), what is the probability that of the first n seeds planted in the farm exactly k germinated?
Answer:
Let p(k, n) be the probability that exactly k of the first n seeds planted in the farm germinated. Using induction on n,we will show that $p(k,n)=\frac{1}{(n−1)}$ for all k< n.
For n=2, $p(1,2)=1=\frac{1}{(2−1)}$ is true. If $p(k,n−1)=\frac{1}{(n−2)}$ for all k< n-1 then, by the law of total probability,
$$\begin{align}p(k,n)&= \frac{k-1}{n-1}p(k-1,n-1)+\frac{n-k-1}{n-1}p(k,n-1)\\&=\frac{k-1}{n-1}\frac{1}{n-2}+\frac{n-k-1}{n-1}\frac{1}{n-2}=\frac{1}{n-1}\end{align}$$
I would like to see why the formula is true without trying out on a few small cases and seeing that they are all $\frac{1}{n-1}$ and then verifying by induction.