Find the probability that in 200 tosses of a die you get exactly 30 fives?

Question

Find probability that in 200 tosses of a fair die, we will obtain exactly 30 fives

Answer 1

In 200 tosses of a die we are require 30 fives and 170 non-fives. The number of ways of selecting 30 results out of 200 is very high: 4.1E35. This the relevant coefficient in the binomial expansion of (p+(1-p))^200 where p=1/6, the probability of tossing a 5 and 1-p=5/6, the probability of tossing a different number. So we have the expression 4.1E35*p^30*(1-p)^170 as the probability of tossing exactly 30 fives. This comes to 0.0641 or 6.41%.