Bill and Jane play a simple game involving two fair dice, each of

Consider the case of Bill
He can have 6C1*6C1= 36 possibilities for the throw.

Same for Jane, She can have 6C1*6C1= 36 possibilities for the throw.

Thus total possible cases= 36*36= 1296

Number of ways both Bill and Jane get a 2:
Bill: (1,1) and Jane: (1,1) = 1
This is same for the case when the both get 12:
Bill: (6,6) and Jane: (6,6) = 1

Number of ways both Bill and Jane get a 3:
Bill: (1,2|2,1) and Jane: (1,2|2,1) = 2C1*2C1= 4
This is same for the case when the both get 11:
Bill: (5,6|6,5) and Jane: (5,6|6,5) = 4

Number of ways both Bill and Jane get a 4:
Bill: (1,3|3,1|2,2) and Jane: (1,3|3,1|2,2) = 3C1*3C1= 9
This is same for the case when the both get 10:
Bill: (4,6|6,4|5,5) and Jane: (4,6|6,4|5,5) = 9

Number of ways both Bill and Jane get a 5:
Bill: (1,4|4,1|2,3|3,2) and Jane: (1,4|4,1|2,3|3,2) = 4C1*4C1= 16
This is same for the case when the both get 9:
Bill: (3,6|6,3|5,4|4,5) and Jane: (3,6|6,3|5,4|4,5) = 16

Number of ways both Bill and Jane get a 6:
Bill: (2,4|4,2|3,3|1,5|5,1) and Jane: (2,4|4,2|3,3|1,5|5,1) = 5C1*5C1= 25
This is same for the case when the both get 8:
Bill: (2,6|6,2|5,3|3,5|4,4) and Jane: (2,6|6,2|5,3|3,5|4,4) = 25

Number of ways both Bill and Jane get a 7:
Bill: (3,4|4,3|2,5|5,2|6,1|1,6) and Jane: (3,4|4,3|2,5|5,2|6,1|1,6) = 6C1*6C1= 36

Total number of ways for draw:
2*1 + 2*4 + 2*9 + 2*16 + 2*25 + 36= 146

Thus total ways in which either Bill or Jane wins:
1296-146= 1150

Jane will win in half the cases whereas Bill will win in half the cases
Hence Jane wins in 1150/2= 575 cases

Thus total probability
575/1296

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Hi Bunnel

Can you please explain why the answer is not 1/2 here ?
If we try to solve logically when both have fair chance of winning here why the probability of winning for each is not 1/2 ? Both have equal chance of winning here.

Hi Radhika11,

According to the prompt, Jane only 'wins' if she scores HIGHER than Bill. She does NOT win if she TIES his score. While they would each have an equal chance of outscoring the other, the probability would NOT be 50/50.

GMAT assassins aren't born, they're made,
Rich

isn't there a faster method to calculate the answer in this case?
p.s. it took me 5 minutes to get the correct answer

I am still unable to find a faster way of calculating the probability of a draw. All of the methods suggested above involve a lot of calculation and I strongly believe require more than 2 minutes. Bunuel do you have any suggestions ?

GMATinsight ; sir is there any easy/ short way to solve this question ? i am honestly blown away with the count of cases involved to solve this question ..

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