Question

What is the probability of rolling a total of 7 with two dice at least once in 10 rolls?

Answer 1

`P("at least one 7 in 10 rolls of 2 dice")~~83.85%`

Explanation:

When rolling 2 dice there are 36 possible outcomes.
[to see this imagine one die is red and the other green; there are 6 possible outcomes for the red die and for each of these red outcomes there are 6 possible green outcomes].

Of the 36 possible outcomes 6 have a total of 7:
`{color(red)1+color(green)6,color(red)2+color(green)5,color(red)3+color(green)4,color(red)4+color(green)3,color(red)5+color(green)2,color(red)6+color(green)1}`

That is `30` out of `36` outcomes will not be a 7 total.
`3/36=5/6`

We will not get a total of `7` on the first roll `5/6` of the time.

Of the `5/6` of the time we did not get a `7` on the first roll,
we will not get a `7` on the second roll `5/6` of the time.
That is `5/6xx5/6=(5/6)^2` of the time we will not get a total of `7` on either of he first two rolls.

Continuing with this reasoning, we see that we will not get a total of `7` on any of the first `10` rolls `(5/6)^10` of the time.

With the help of a calculator we find that we will not get a total of `7` on any of the first `10` rolls approximately `16.15%` of the time.

This implies that we will get a total of `7` on at least one of the first `10` rolls `100%-16.15%=83.85%` of the time.