Question

You roll eight six-sided dice. What is the chance that you rolled exactly two fives? (0) 0 6 6

Answer 1

The correct option is the third option: $\boldsymbol{\binom{8}{2} \left(\frac{1}{6} \right )^2\left(\frac{5}{6} \right )^6}$ [ANSWER]

Explanation:

Consider a six-sided die. There are six faces out of which only one face has the number 5. Thus, the probability of rolling a 5 on a roll of a six-sided die is given by:

P(rolling a 5) = (Number of faces with number 5) / (Total number of faces) = 1/6

Now, we are given that we roll eight six-sided dice, let X denote the number of fives rolled.

Now, since there is a fixed number of dice (equal to 8), each die has two outcome (rolling a 5 and not rolling a 5) and each die rolls a 5 with probability 1/6 independently of other dice, thus we can conclude that:
X ~ Binomial(n = 8, p = 1/6) and the probability mass function of X is given by:

$\begin{align*} P(X=x) &= \binom{n}{x}p^x(1-p)^{n-x} \ \ \ \ \ \ \ \ \ \ \ &&;x=0,1,2,...,n \\ &= \binom{8}{x}\left(\frac{1}{6} \right )^x\left(\frac{5}{6} \right )^{8-x} \ \ \ \ \ \ \ \ \ \ \ &&;x=0,1,2,...,8 \end{align*}$

Thus, the probability or chance that we rolled exactly two fives is given by:

$\begin{align*} \bf P(X=2) &= \binom{8}{2}\left(\frac{1}{6} \right )^2\left(\frac{5}{6} \right )^{8-2} \\ &= \bf \binom{8}{2}\left(\frac{1}{6} \right )^2\left(\frac{5}{6} \right )^{6} \ \ \ \ \ \ \ \ \ \ \ \ [ANSWER] \end{align*}$

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