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?

1 Answer
marfre
Mar 15, 2018

#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#

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