Pileup Probability

The purpose of this exercise is find an expression for pileup probability (Pin). That is, given a hit area density r what is the probability that one or more hits lands in an area "a".

Giving the bottom line before the derivation:

Pileup probability in:

area

hit density

The derivation is done a little backward by noting that

where Pout is the probability that no hit is found in the area "a"

We derive an expression for

Consider an area "a" inside a larger area A, now throw a stone n times at random into A, then the probability that it never lands in "a" is given by:

eq. 1

The number of throws, n and the size of A are related to by the specified hit density:

eq. 2

So, we can make n and A arbitrarily large and will do that eventually.

eq. 3

Using the binomial theorem

express the sum starting with k=n and stepping k down gives

.....

rewriting this a little

. . . .

eq. 4

So, using eq. 2 we can rewrite p as:

and substituting into eq. 4 gives:

. . . . eq. 5

For n large consider each term in the sum (eq. 5), expressions in n in the numerator are only very slightly less than expressions in n in the denominator, so in the limit of large n we can write eq. 4 as:

. . . .

eq. 6

Note, eq. 6 is now the same as the series expansion of the exponential function

So what we started with in eq. 1 is now

But what we are really interested in is Pin, the probability that one or more hits is in the area "a" and this is just:

eq. 7

The desired result.

Check the expression, for n less than

from eq. 5

from eq. 6

Compare the two expressions

The discrepancy is less than 4% for N = 100 and becomes less as N increases.