Probability of throwing exactly 100 sixes on an unfair dice.

Question

Suppose we have a dice with probabilities $\frac{1}{6}$ for rolling one, $\frac{1}{12}$ for rolling two, $\frac{1}{12}$ for three, $\frac{1}{6}$ for four, $\frac{1}{6}$ for five, and $\frac{1}{3}$ for six. What is the probability of rolling exactly 100 sixes from 250 rolls? I think I should use multinomial distribition in this case. My try: C(150,250)*(2/3)^150/6^250

Answer 1

This is not a multinomial distribution but a binomial distribution. Yes, when you roll $250$ dice you'll probably get a mixture of ones, twos, threes, fours, fives, and sixes, but we're only interested in a success which happens when $X=6$ and a failure when $X\in\{1,2,3,4,5\}$

Use the standard binomial:

$$P(X=k)={n \choose k}p^k(1-p)^{n-k}$$

where $$n=250$$

$$k=100$$

$$p=\frac{1}{3}$$