A fair die is rolled 3 times. A 4 is considered "success", while all other outcomes are "failures". What is the probability of 2 successes?

Mar 1, 2017

$P \left(\text{2 successes}\right) = \frac{5}{72}$

Explanation:

For any particular throe of the dice, then:

 P(S) = 1/6 ; P(F) = 5/6

So if we roll the dies three times; then the permutations are as follows;

Thus the probably of 2 successes is:

$P \left(\text{2 successes}\right) = P \left(F S S\right) + P \left(S F S\right) + \left(S S F\right)$
$\text{ } = \frac{5}{6} \frac{1}{6} \frac{1}{6} + \frac{1}{6} \frac{5}{6} \frac{1}{6} + \frac{1}{6} \frac{1}{6} \frac{5}{6}$
$\text{ } = 3 \cdot \frac{5}{6} \frac{1}{6} \frac{1}{6}$
$\text{ } = \frac{15}{216}$
$\text{ } = \frac{5}{72}$