Answer to Question #95988 in Statistics and Probability for anna

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