Question

If you roll five dice, what are the odds of rolling five 5's?

Answer 1

Hmm..

Explanation:

1 Dice: chance of 1 in 6, `rarr 1/6.`

2 dice: chance of 1 in 6 for each, so: `rarr 1/6 . 1/6 = 1/36`

For 2 dice: 1 in `6^2` = 1 in 36;

Therefore for 5 dice: 1 in `6^5` = 1 in 776 ...

Answer 2

The odds in favour are `1:7775`.

Explanation:

Assuming all five dice outcomes are independent, we can rephrase the question as, "If we roll one die five times, what is the probability of getting a 5 on each roll?"

Since there is one "successful" outcome (rolling a 5) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6), the probability of rolling a 5 on one roll is `P("roll one 5")=1/6.`

Here's where the independence comes in. Since we're treating the 5 dice as independent, the probability of rolling five 5's is the same as the probability of rolling one 5, five times. As such, we can multiply `P("roll one 5")` by itself 5 times, and it will give us the same answer as `P("roll five 5's").`

`P("roll five 5's")= [P("roll one 5")]^5`
`color(white)(P("roll five 5's"))= (1/6)^5`
`color(white)(P("roll five 5's"))= 1/7776`

Finally, since this question asked for the odds of rolling five 5's (and not the probability of it), we take the ratio `(P("rolling five 5's"))/(P("NOT rolling five 5's"))` which gives

`(1//7776)/(7775//7776)=1/7775`

Since this is an odds value, it is often written as `1:7775`. This says that for every 1 success, we should expect 7775 failures.