After rolling four dice, what is the probability that all four dice show different numbers?

Oct 17, 2017

$\text{P} \left(E\right) = \frac{5}{18}$

Explanation:

Let $E$ be the event 'All four dice show different numbers' and $S$ be the sample space of the experiment. Then $\left\mid S \right\mid = {6}^{4}$ and $\text{P"(E)=absE"/} \left\mid S \right\mid .$

In order to find $\left\mid E \right\mid$, we can consider the four dice rolls independently. On the first roll, there is no risk of duplicating any previous roll (since there aren't any), so there are 6 acceptable outcomes. For the second roll, we want to avoid whatever the first roll was, so there are 5 acceptable outcomes.

If we keep going, assuming we keep rolling different numbers, then on roll three, there are 4 acceptable outcomes, and on roll four, there are 3 acceptable outcomes. (As the roll counter goes up, the number of possible unique rolls goes down.)

Assuming independence of rolls, we get $\left\mid E \right\mid = 6 \times 5 \times 4 \times 3 ,$ which gives $\left\mid E \right\mid = 360.$

Since $\text{P} \left(E\right) = \frac{\left\mid E \right\mid}{\left\mid S \right\mid}$, we get

$\text{P} \left(E\right) = \frac{360}{6} ^ 4 = \frac{60}{216} = \frac{5}{18} ,$

which is approximately 27.78%.