Suppose there is a line of length $L$ cm. And it begins to rain at a constant rate of one droplet per second. Once a drop strike the line and it wets 1 cm of the line. What is the expected number of droplets it takes to wet the whole line?
The following condition was suggested by Henry:
To avoid the boundary problem, each point on the line within a distance of $\frac{1}{2}$ the point struck is wetted. The centre of the drop can be anywhere on the line (so if near the end would cover less than 1 but at least $\frac{1}{2}$ cm of the line.
The discrete version of this problem is a classical coupon collector problem，I am curious what would happen in the continuous case. The related post of this problem is rain droplets falling in a table.
Any help will be appreciated. Thanks in advance.
Your problem is equivalent to breaking a stick of length $1$ into $n$ pieces1, and then asking what the expected number of $n$ is such that every piece has length $\leq \frac{1}{L}$. This problem is studied in "On the lengths of the pieces of a stick broken at random" by Lars Holst.
The exact chance of that happening after breaking the stick into $n$ pieces is:
$$\sum_{v=0}^n (-1)^v\binom{n}{v}\max\left(0, 1 - \frac{v}{L}\right)^{n-1}$$
They also proved that if it takes $N$ pieces such that each piece $\leq \frac{1}{L}$, then
$$P\left(\frac{N}{L} - \ln(L) - \ln(\ln(L)) \leq x\right) \rightarrow \exp(-e^{-x})$$
So $N \approx L(\ln(L) + \ln(\ln(L))$.
1To handle the edges you can add $\frac{1}{2}$ to both sides of the line. So, effectively, my whole answer above should use $L' = L + 1$.