Bill and Jane play a simple game involving two fair dice, each of which has six 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 choice from the following options:  A) 15/36  B) 175/432  C) 575/1296  D) 583/1296  E) 1/2
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 - 2,3,4,5,6,7,8,9,10,11,12   Prob. of each sum is -   2 - 1,1 - 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. 575/1296
<result>C</result>
Bill and Jane play a simple game involving two fair dice, each of which has six 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?