The probability of tossing a 5 is ⅙. So the probability of not tossing a 5 is ⅚.
Let p=⅙ and q=⅚.
In 200 trials we can use the binomial expansion:
(p+q)200=1=p200+200p199q+...+200C30p30q170+...+200pq199+q200, which represents:
(probability of all 5s)+(probability of 199 5s)+...+(probability of 30 5s)+...
200C30=4.0968×1035 is the number of combinations of 30 5s and 170 different dice numbers.
p30q170=⅙30⅚170=5170/6200=1.5655×10-37.
So the probability of throwing 30 5s in 200 separate throws is:
4.0968×1035×1.5655×10-37=0.064135=6.4135% approx.
Tables give the same result.