What is the probability of collecting all 50 state quarters from a bucket of 1000 of them based on known probabilities of each zone of origin? I live in the Eastern time zone of USA.  Assume that there are 5 "zones" the USA is divided into for this problem (Eastern, Central, Mountain, Pacific, and Alaska + Hawaii).  Also assume that if the probability of finding a distinct (needed) quarter in my own (Eastern) zone is p (for the first one drawn from the bucket of 1000), then the other zones would be p/2, p/3, p/4, and p/30 respectively (also on the first draw from the bucket of 1000).  For example, the probability of me finding a needed quarter in my own time zone (Eastern) is 4 times as great as me finding one from one from the Pacific zone.  Another way of looking at it is it is 4 times as difficult for me to find a needed quarter from CA than it is from Florida (FL) for example.  Also note that "bad" quarters (not needed) will NOT go back into the main bucket, but rather into a 2nd bucket of "repeats" which we will never draw from.  For simplicity, assume there are 12 states in each of the 4 main zones except for the 5th zone which is only Alaska and Hawaii.  I separated those 2 states since they are non-continental, thus I assigned them a higher difficulty than any of the other 48 states.
So assuming I have a bucket of 1000 normally circulated state quarters (none of the old style "non-state" quarters), what is the probability I will get all 50 states "covered" in that bucket of 1000 quarters?  That is, at least one of each state quarter.
As a bonus question, what is the probability I can fill 2 coin albums with all 50 state quarters from that same bucket of 1000 (100 "good" quarters needed total).
Reason I ask is I will actually be doing this and was curious how to compute this probability mathematically.
 A: You can compute the chance of each state by choosing $p$ so the probabilities add to $1$.  Let the probabilities of each state be $q_i$ where $i$ runs from $1$ to $50$ and the states are in some order.  We can make a slight approximation by assuming whether you get each state is independent.  This is not quite true, because if you have at least two New York quarters you have slightly less chance to have two Vermont quarters, but it will not be far off.  The chance you do not get state $i$ is 
$\left(1-q_i\right)^{1000}$ so the chance you do get state $i$ is $1-\left(1-q_i\right)^{1000}$.  The chance you have all $50$ states is then 
$$\prod_{i=1}^{50}1-\left(1-q_i\right)^{1000}$$
The chance you get exactly one quarter of state $i$ is $1000q_i\left(1-q_i\right)^{999}$ from the binomial distribution, so the chance you have at least two of state $i$ is $1-\left(1-q_i\right)^{1000}-1000q_i\left(1-q_i\right)^{999}$.  The chance you have at least two of each state is then $$\prod_{i=1}^{50}1-\left(1-q_i\right)^{1000}-1000q_i\left(1-q_i\right)^{999}$$ 
With $12$ states in each of the first four time zones, we solve $12p(1+\frac 12+\frac 13+\frac 14)+\frac {2p}5=1$ and have $p=\frac 5{127}$, so $q_i=\frac5{127}$ for $12$ states, $\frac 5{254}$ for $12$ more, and $\frac 1{127}$ for Alaska and Hawaii.  I make it about $0.99863$ chance of having at least one of each state and $0.98653$ chance of having at least two of each state.  Here the expected number of Alaska and Hawaii quarters is almost $8$ each, so you really expect to have two of each.
A: I will simulate this on a computer to check the mathematical results. To generate a "weighted random" bucket of 1000 quarters, I will use the ratios of $72,000 : 36,000 : 24,000 : 18,000 : 2,400$ for the 5 zones respectively.  Note that those are all divisible by 12 and it allows for all 1000 quarters to be from any single state, just as it could be in real life, although extremely unlikely.  So the correct expected values will be retained. I will just put these in an array and choose 1000 unique random numbers from 1..152,400 inclusive, and then add each of those quarters to the bucket, then see if I can fill all 50 states from them (perhaps even retaining the average # of quarters used to make this happen).  I will do thousands of simulations to get a good average and will report back the results maybe in a few days.
