Selections From
Multiple random selections from a population…?
I am looking for a general formula that would allow me to answer the following type of question:
I have a population of 1000 people. I give 85 of them (selected randomly) a penny. Then, from the 1000 I randomly select 32 people and give them a penny (so there might be some overlap). Then, I select 422 randomly and give them a penny, etc.
When all this is done, now many people have no pennies, how many have one penny, how many have two pennies, how many have 3 pennies, etc.
Nomen est omen.
We have n rounds of distribution of pennies. At each round each person gets either 1 or 0 pennies.
The chances of finding a person having 1 penny of round i is a(i), not having 1 is 1- a(i).
After n rounds of distribution, the maximum number of pennies 1 person can have is n the minimum is 0. The chance of having no pennies = product over all (1-a(i)) we call this P(0) = prod(a(i),0)
The chance that 1 person has got 1 penny in a specific round (k) is product over all (1-a(i)) in which (1-a(k) is replaced by a(k). So the chance in 1 penny overall is the sum of all n
P(1) = sum(prod(a(i),1)))
Here prod(a(i),1)) means product of all a(i) with in every product a a(i) replaced by (1-a(i)).
The chance that 1 person has 2 pennies is specific round k and is the product over all (1-a(i)) in which (1-a(k)) is replaced by a(k) and (1-a(l)) is replaced by a(l). We call this product prod(a(i),2)). When we sum over all k <> l we get each product twice so we have to divide by 2.
P(2) = (1/2).sum(prod(a(i),2)))
The chance 1 person has 3 pennies is likewise. Instead of diving by to we have all permutations of the 3 choices so (n over x) is x is the number of pennies.
So the chance of 1 person having x pennies is P(x) = sum(prod(a(i),x)))/(n over x)
prod(a(i),x) = product of all (1-a(i)) with x terms changed form (1-a(k)) to a(k)
As all a(i) can be different, there is no chance on simplifying this for the general case.
Leonard Bernstein – Candide (1956) – Selections from the Original Broadway Cast Recording
