bill and jane play a simple game involving two fair dice , each of which has 6 sides numbered from 1 to 6 ( with an equal chance of landing on any side ) . bill rolls the dice and his score is the total of the two dice . jane then rolls the dice and her score is the total of her two dice . if jane ’ s score is higher than bill ’ s , she wins the game . what is the probability the jane will win the game ?
Choose the correct answer.
A) 15 / 36
B) 175 / 432
C) 575 / 1296
D) 583 / 1296
E) 1 / 2

Rationale: there are three possible outcomes in the game : 1 . b wins 2 . j wins 3 . draw so , 1 = p ( j ) + p ( b ) + p ( draw ) 1 = p ( j ) + p ( b ) + p ( draw ) the probability of b winning or j winning the game is same . so , 1 = 2 ∗ p ( j ) + p ( draw ) 1 = 2 ∗ p ( j ) + p ( draw ) so , we need to calculate the probability of a draw . a draw can happen when both will have the same sum . the possible sums are - 2 , 34 , 56 , 78 , 910 , 1112 prob . of each sum is - 2 - 11 - 1 / 36 3 - 1 2 , 2 1 - 2 / 36 4 - 1 3 , 2 2 , 3 1 - 3 / 36 5 - 1 4 , 2 3 , 3 2 , 41 - 4 / 36 6 - 1 5 , 2 4 , 3 3 , 4 2 , 5 1 - 5 / 36 7 - 1 6 , 2 5 , 3 4 , 4 3 , 5 2 , 61 - 6 / 36 8 - 2 6 , 3 5 , 4 4 , 5 3 , 6 2 , - 5 / 36 9 - 3 6 , 4 5 , 5 4 , 63 - 4 / 36 10 - 4 6 , 5 5 , 6 4 - 3 / 36 11 - 5 6 , 6 5 - 2 / 36 12 - 6 6 - 1 / 36 now , since both have to get the same sum , the probability will be same for both and the combined probability will be the square of the probabilities i . e . 2 - ( 1 / 36 ) ^ 2 3 - ( 2 / 36 ) ^ 2 etc . squaring the probabilities and adding them will give 146 / 1296 so , p ( d ) = 146 / 1296 1 = 2 * p ( j ) + p ( d ) 2 * p ( j ) = 1150 / 1296 p ( j ) = 575 / 1296 ans . c
Answer: C