Suppose I want to randomly select 3 integers between 1 and 100 with replacement - for example (34, 93, 3) or (2,89,2). I want to know the probability that these 3 numbers will sum to some specified number, e.g. 50.
Since the probability of obtaining any combination of 3 numbers is equal - I know that this problem takes the general form of : Probability = number of valid ways / number of total ways.
From this post (Number of ways of choosing $m$ objects with replacement from $n$ objects), I know that the number of total ways that 3 numbers with replacement (out of 100) can be picked is : 100!/((3!) * (97!)) = 161700 ways
However, I am not sure how to calculate the total number of "valid" ways in which 3 random integers between 1-100 can sum to 50.
Had this been a simpler problem, e.g. 3 random integers (with replacement) between 1 and 10 must sum to 7 - I could have manually enumerated all "valid" ways.
But in this problem, is there some standard formula that can be used to calculate the number of "valid" ways that 3 random integers between 1 and 100 can sum to 50?
The first thought I had was to attempt to calculate this number using Markov Chains. Additionally, a Markov Chain would also allow me to find out the average number of times I would need to keep picking 3 random numbers until they summed to 100 ("mean time to absorption") - but this Markov Chain would have so many states, I would have no idea how to write the transition matrix for such a Markov Chain.
Can someone please show me how to calculate these numbers (i.e. the probability that 3 random numbers with replacement sum to 50, and the number of times 3 random numbers need to be selected with replacement until the first triplet sums to 50) - preferably using Markov Chains?