Question

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

Answer 1

`"P"(E)=5/18`

Explanation:

Let `E` be the event 'All four dice show different numbers' and `S` be the sample space of the experiment. Then `absS=6^4` and `"P"(E)=absE"/"absS.`

In order to find `absE`, 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 `absE=6 xx 5 xx 4 xx 3,` which gives `absE=360.`

Since `"P"(E)=absE/absS`, we get

`"P"(E)=360/6^4=60/216=5/18,`

which is approximately 27.78%.