Mathematical Biology.
This week you are going to try to derive an equation used in dynamical systems in mathematical-biology.
To derive this equation, imagine
that there is a board of area
. Furthermore on this board there are randomly placed discs. Each
disc has an area of
.
There are 
discs and this is kept at a constant. Next, we have a
machine that taps the board on random spots with a pointer at k
taps per unit of time t. If the machine taps a disc then it will
spend
units
of time “capturing” the disc off the board before it
starts tapping again (at this time another disc is randomly placed
on the board to keep the number of discs constant).
Find the expected value of the rate of capture of discs.
HINT:
Find the probability that the machine taps a disc. Think about the possible outcomes of a tap hitting a disc and how they relate to the area of the board and discs. If we sum the probabilities of success for each tap in the given time interval and isolate a certain term we will get the expected number of discs captured. Upon dividing by t you will get the expected rate of capture.