Question

What are the odds and probability of rolling 5 dice, and all 5 dice are the same number?

Answer 1

`1/6^4=1/1296`

Explanation:

There are six faxes with six distinct numbers. For each dice, the
probability is (number of favorable cases)/(the total number of all possible cases).
Because any first roll is favorable, this probability can be modeled by:
`=6*1/(6^(n)` or `1/(6^(n-1))`

For all showing the same number, it is the compound probability
= product of the separate probabilities.
`=1/(6^(5-1))`
`=1/6^4`

Answer 2

`P("all 5 dice show the same number") = 1/1296`

Odds are `1:625`

Explanation:

Let's look at the probability first.

`P(5 " dice give the same number")`

They must be all `1s` or all `2s` or all `3s` ... and so on

`=P(1,1,1,1,1) +P(2,2,2,2,2)+......... + P(6,6,6,6,6)`

`(1/6xx1/6xx1/6xx1/6xx1/6) + ....+(1/6xx1/6xx1/6xx1/6xx1/6)`

`= 1/6^5 + 1/6^5 + .......... +1/6^5`

`= 6 xx 1/6^5`

`=1/6^4 = 1/1296`

The odds are given as a ratio of the number of ways this will happen compared to the number of ways it doesn't.

So if the first die is `1`, the other four will not be,

`6 xx (1/6 xx5/6xx5/6xx5/6xx5/6) = 625/1296`

The odds for this happening are `1 : 625`