I am a bioinformatician with a probability calculation problem. It is seemingly trivial, but I get lost very quickly with probibilities.
I need to infer a calculus, a formula, with which I can assess the probability, no matter how small or big, that two urns with balls give disjoint sets of outcomes by design.
Lets say that we have 2 urns, where there are steel balls and teflon balls. The sampling device (with replacement) is magnetic so the teflon balls are never drawn.
Our urns have exactly 20 balls each, and each ball is represented only once, so our theoretical set of outcomes is 1-20.
So, I know that there are some teflon balls (not drawable) in my 2 urns. I am suspecting that the remaining sets of steel balls are disjoint between the 2 urns.
A working example:
Urn A, 20 draws with replacement, [1,2,3,4,5,6,7,8,9,10] the outcomes.
Urn B, 20 draws with replacement, [13,14,15,16,17,18,19,20] the outcomes.
Can I calculate the probability, no matter how small or big,
that in urn A, balls [13-20] are in teflon, given that they were not drawn in the first 20 trials,
that in urn B, balls [1-10] are in teflon, given that they were not drawn in the first 20 trials.
In other words, a theoretical calculation of the eventuality that the two urns can only have disjoint outcomes - no matter how small or big the probability of that.
Please let me know if my description is poor, in which case I can rephrase.
Thank you very much in advance.