So I new to probability and need someone to help me out if you could. I wanted to look into the probability of a process being complete if each operation of the process has its own likely hood of success or failure. I know that I should be using a binomial distribution to study the process. Lets say I have 3 operations (A, B, C) that need to be completed in a process. Each operation determines the number of parts that move to the next operations, but the probability of the parts being processed correctly at each operations is independent of the previous operations. So because they are independent (I'm assuming) then... ##P(A \cap B \cap C) = P(A)P(B)P(C)## This is where I am having issues. The number of parts (n) moving to the next operation is dependent on the last operations but the probability (p) for each operation isn't dependent on the last operation or the next operation. If my last assumption is true then I want to take the product of all three to find the distribution for the whole process. If I trial n parts through the entire process. ##P(A \cap B \cap C) = BINOMDIST(k,n,p_A)BINOMDIST(k,n,p_B)BINOMDIST(k,n,p_C)## I need some help with the mode of this product. All I can think of doing is taking the derivative with respect to k and setting it to zero to solve for the mode, but this being discrete I know this is not correct. I have looked for a derivation for the mode of the Binomial Distribution online, but all I find is the solution not the steps to get there. I think if I saw the whole derivation of it I could handle the mode of the product. Also, do powers of the binomial coefficients simplify. This looks like it could get messy. Thanks for any help you can offer.