Your question is not well posed. If you are asking about throwing n dice
triplets, where each triplet is uniquely identified, then you have already
explained why the number of triple 6's would be Binomial (n,1/216).
Otherwise you need to state more clearly just what you are asking.

Jerry

"Myles" wrote:

I am working on a project which is loaded with probability outcomes. To
parody a segment of my problem, consider throwing 3 dice (each numbered
1 through 6 as usual). The probability of scoring 3 sixes with one
throw (termed a success) = (1/6)^3 or 1/216. If this exercise is
repeated n times, there is a greater chance to score a success but
there is yet a chance of not scoring (a success) at all, regardless of
the value of n. It would sound intuitive that when n=216, at least one
success should be registered but this is most certainly not the case as
the law of averages fail here.

Now, can someone compute, using bimomial expansion or otherwise, the
statistical probability of at least scoring one set of 3 simultanoeus
sixes throwing all 3 dice at any one time? What value of n (or limit
thereof) attaches to this outcome?