Hi. I'm working on deriving a probability formula to help in some research I'm doing, but I've hit a wall. I've managed to reduce my problem down to a dice problem, which effectively makes it a textboot problem. There one aspect of the problem I think I've solved, but another that I can't seem to crack.
Part I understand: If I roll 8, 4-sided dice, what is the probability that I will get 4 unique pairs? (ie, 1 1 2 2 3 3 4 4)
My solution: (8 2)(6 2)(4 2)(2 2) / 4^8
[ (8 2) is 8 choose 2 ]
That one seems to work, and in the end simplifies down to a multinomial coefficient. (Which makes sense.)
Now, I expand the problem: If I roll 16, 4 sided dice, what is the probability that I will get 4 unique pairs? (I don't care what the rest of the dice do, but at minimum I need the 4 pairs. So, I could get 1 1 2 2 3 3 4 4 1 2 3 4 4 4 3 and be happy.)
Originally I thought it would be (16 2)(14 2)(12 2)(10 2) / 4^16, but I think I've now realized that that dice I don't care about are causing me trouble.
Does anyone have any thoughts?
Part I understand: If I roll 8, 4-sided dice, what is the probability that I will get 4 unique pairs? (ie, 1 1 2 2 3 3 4 4)
My solution: (8 2)(6 2)(4 2)(2 2) / 4^8
[ (8 2) is 8 choose 2 ]
That one seems to work, and in the end simplifies down to a multinomial coefficient. (Which makes sense.)
Now, I expand the problem: If I roll 16, 4 sided dice, what is the probability that I will get 4 unique pairs? (I don't care what the rest of the dice do, but at minimum I need the 4 pairs. So, I could get 1 1 2 2 3 3 4 4 1 2 3 4 4 4 3 and be happy.)
Originally I thought it would be (16 2)(14 2)(12 2)(10 2) / 4^16, but I think I've now realized that that dice I don't care about are causing me trouble.
Does anyone have any thoughts?