Question

You and your friend play a game in which you roll two dice. The first person to roll two dice with a sum of 5 wins. If you roll first, what is the probability that you win?

Answer 1

`9/17`

Explanation:

Given: Two people play until you win with `2` dice, roll a sum of `5`. You are the first to play.

You can create a table of possible rolls. Each dice can roll numbers `1 - 6`. There are a possibility of `6 *6 = 36` different rolls

`ul(" "1" "2" "3" "4" "5" "6" ")`
`" "1 | " "1,1|" "1,2|" "1,3|" "color(red)(1,4)|" "1,5|" "1,6`
`" "2 | " "2,1|" "2,2|" "color(red)(2,3)|" "2,4|" "2,5|" "2,6`
`" "3 | " "3,1|" "color(red)(3,2)|" "3,3|" "3,4|" "3,5|" "3,6`
`" "4 | " "color(red)(4,1)|" "4,2|" "4,3|" "4,4|" "4,5|" "4,6`
`" "5 | " "5,1|" "5,2|" "5,3|" "5,4|" "5,5|" "5,6`
`" "6 | " "6,1|" "6,2|" "6,3|" "6,4|" "6,5|" "6,6`

The probability you win: `P("sum"=5) = 4/36 = 1/9`

The probability that you fail is `1- 1/9 = 8/9`

The probability that you win in any number of moves is:
`p = 1/9 * 1 + 8/9 (1-p)`

Solve for `p`:
`p = 1/9 + 8/9 - 8/9p`

`p = (9 - 8p)/9`

`9p = 9 - 8p`

`17p = 9`

`p = 9/17`