Poisson vs binomial process

[zezima1]

I might need you guys to help me see how this proces, will be distributed:

Suppose we have a large amount of elements N(≈10¹²). I'm simulating a system where I for each iteration damage a random element. If an element gets damaged its damagecounter goes up 1.
So say I pick element number 100123 on the first iteration. Its damagecounter will now count +1 whilst the damagecounters of the other elements in the system will be 0.

I want to write an expression for the probability that a specific element acquires k damages in t iterations. Here are my steps in deriving an expression:

1) The probability for a specific element to get damaged must be 1/N.

2) Thus the probability that a specific element gets k damages after t iterations must be:

p(x=k) = K(t,k) * (1/N)^k * (1-1/N)^t-k

where K(t,k) denotes the binomial coefficient.

Now, I want to find the total probability for an element to have acquired k damages during our t iterations. Thus I multiply by N and end up with:

p(x=k) = N * K(t,k) * (1/N)^k * (1-1/N)^t-k

Now first of all, I want to know: Would this expression be correct or am I making any mistakes/illegal assumptions in the derivation?

Second, my model appears to follow a poisson distribution, when I plot the amount of elements with damage 1->10 versus time. Does that fit with this theory? I can't quite see if that is a good thing or not?

 

[mathman]

I won't comment on your derivation.

However, in general, Poisson distribution is a good approximation for the binomial for large N and Np fixed.