The water level of a certain reservoir is depleted at a constant rate of $1000$ units daily. The reservoir is refilled by randomly occurring rainfalls. Rainfalls occur according to a Poisson process with rate $0.2$ per day. The amount of water added to the reservoir by a rainfall is $5000$ units with probability $0.8$ or $8000$ units with probability $0.2$. The present water level is just slightly below $5000$ units.
What is the probability the reservoir will be empty sometime within the next ten days?
Edit: The answer is : $$e^{-1}+e^{-1}\left(0.8\right)e^{-1}$$
After 5 days it would be just empty with no rainfall, P = $e^{-1}$
if you have just one 5000 in the first 5 days,
prob $P = 0.8e^{-1}$
then you are back in the same situation as at time zero (in all other cases you are either emptied or have sufficent water for the next 5 days in all cases). At 5 days, with 5000 rainfall, you also lost 5000 rainfall, exceeding what you started with, so you'd have slightly less than 5000 again. 10,000 or 8,000 will always see you through.
so it comes out as
P(empty) = P(empty at 5 days) + P(1 x 5000 after 5 days) x P(empties in 5 days)
= $e^{-1} + 0.8e^{-1}e^{-1}$
it just so happens that he can only become empty at exactly 5 or exactly 10 days, with the latter involving exactly 1 of 5000 rainfall