# Combining Multiple Binomial (or Negative Binomial) Distributions

#### bobryuu

I have been struggling with a problem, and am seeking help on the topic:

I have two dice, one with a faces and the other with b faces. Only one success can come from each die, such that the success rates of the dice are 1/a and 1/b. I want to know the expected number of times I must throw both dice to such that I obtain an n number of successes.

I know from the means of both the binomial distribution and the negative binomial distribution that I must roll die a an average of a*n times to get n successes and very similar for die b, but I can't seem to figure out how to combine the probabilities to get the answer.

#### bobryuu

I've come a little bit closer to the answer on this topic. I've turned it into a Multinomial Probability problem:

Four Probabilities:
(b-1)/(a*b) => Counts towards the success of die A
(a-1)/(a*b) => Counts towards the success of die B
1/(a*b) => Counts towards the success of both dice
(a*b - a - b +1)/(a*b) => Failure

Edit: I thought I had my numbers correct, but I had exchanged a digit of 6 for a digit of 7, which threw my calculations.

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#### bobryuu

I've been trudging through myself on this, and I have found the means to figure it out. It is going to require a fair amount of counting first, and then some computational mathemagic for the specific case I am looking for.

#### bobryuu

I've successfully figured out this problem myself. If anyone would wish to see the program I've written to deal with this situation, please private message me.