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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?

1 Answer

Mar 1, 2017

$P (2 successes) = \frac{5}{72}$ $P("2 successes") = 5/72$

Explanation:

For any particular throe of the dice, then:

$P (S) = \frac{1}{6}; P (F) = \frac{5}{6}$ $P(S) = 1/6 ; P(F) = 5/6$

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

Thus the probably of 2 successes is:

$P (2 successes) = P (F S S) + P (S F S) + (S S F)$ $P("2 successes") = P(FSS) + P(SFS) + (SSF)$
$= \frac{5}{6} \frac{1}{6} \frac{1}{6} + \frac{1}{6} \frac{5}{6} \frac{1}{6} + \frac{1}{6} \frac{1}{6} \frac{5}{6}$ $" " = 5/6 1/6 1/6 + 1/6 5/6 1/6 + 1/6 1/6 5/6$
$= 3 \cdot \frac{5}{6} \frac{1}{6} \frac{1}{6}$ $" " = 3* 5/6 1/6 1/6$
$= \frac{15}{216}$ $" " = 15/216$
$= \frac{5}{72}$ $" " = 5/72$

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