I am a bioinformatician with a statistical/combinatorial problem - ...a common condition in my field. A bioinformatician can conceive and write the code for very complex problems of biological data, but when it comes to statistics, it is often complete panic in the ranks !
After this short introduction, I will try to rephrase my problem to make it accessible to a larger number of experts:
I have two Sets of 4-digit combination locks.
Set-1 consists of 300 locks.
Each digit in a lock in set-1 accepts only a reduced set of numbers (so the complete range 0-9 is not available):
digit-1: 12345 (5 possible numbers)
digit-2: 3456 (4 possible numbers)
digit-3: 890 (3 possible numbers)
digit-4: 123456 (6 possible numbers)
I create a combination for each one of my 300 Set-1 locks - I don't care if a combination is repeated more than once.
Set-2 consists of 20 locks
Each digit in a lock in set-2 accepts a different reduced set of numbers:
digit-1: 4567 (4 possible numbers)
digit-2: 123 (3 possible numbers)
digit-3: 567890 (6 possible numbers)
digit-4: 45 (2 possible numbers)
I create again a combination for each one of my 20 Set-2 locks and I don't care if a combination is repeated.
My question is:
How can I calculate the probability that all 20 Set-2 locks produce a combination that is never observed in my 300 combinations of Set-1 locks?
I have started digging my way through permutations and combinations, Binomial coefficients, Bernouli theorem, Pascal triangles, etc., etc... but I am not ashamed to admit that I have large gaps in these areas which are not easily filled without consultation with an expert in the field. In other words I cannot apply the theory on my problem...
Thank you very much in advance for reading and for your suggestions.