Question

If you roll this dice 12 times what is the probability that you will roll three 6s? Work out both by hand for the Binomial distribution.

Answer 1

Take three different numbers $a,b,c$ between $1$ and $12$. The probability of obtaining 6 on dice is $\frac{1}{6}$, probability of obtaining other number is $1-\frac{1}{6}=\frac{5}{6}$ . So probability of obtaining $6$ for $a$-th, $b$-th and $c$-th times and obtaining other numbers for other times is $\left(\frac{1}{6}\right)^3\left(\frac{5}{6}\right)^{12-3}=\frac{5^9}{6^{12}}$.

The number of choosing $a, b$ and $c$ is $\binom{12}{3}$, so the probability of obtaining three 6s is $\binom{12}{3}\frac{5^9}{6^{12}}$

Answer: $\binom{12}{3}\frac{5^9}{6^{12}}$