Probability Problem

Question

How to solve this problem quickly? I need more than 5 mins to get the answer. A long calculation....

Source: MGMAT

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?

(A) 15/36
(B) 175/432
(C) 575/1296
(D) 583/1296
(E) 1/2

OA later

Rezinka · Answer

Is OA E?

yuliawati · Answer

No, the OA isn't E.

Rahul@gurome · Answer

yuliawati wrote:How to solve this problem quickly? I need more than 5 mins to get the answer. A long calculation....
Source: MGMAT

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?

(A) 15/36
(B) 175/432
(C) 575/1296
(D) 583/1296
(E) 1/2

OA later

3 possible outcomes:
P(J) = Probability that Jane wins
P(B) = Probability that Bill wins
P(D) = Probability in a draw (each gets the same score)

P(J) + P(B) + P(D) = 1.
Since Jane and Bill each have the same chance of winning, P(J) = P(B).
Thus:
P(J) + P(J) + P(D) = 1
2*P(J) + P(D) = 1
P(J) = [1- P(D)] / 2

So to determine P(J), we need to determine P(D).

The probabilities of getting the different sums display the following symmetry:

P(2) = P(12)
P(3) = P(11)
P(4) = P(10)
P(5) = P(9)
P(6) = P(8)

Let's determine the various probabilities to discern the pattern:
P(2) = P(1,1) = 1/6 * 1/6 = 1/36.
P(3) = P(1,2) + P(2,1) = 1/36 + 1/36 = 2/36
P(4) = P(1,3) + P(3,1) + P(2,2) = 3/36
Now we see the pattern:
P(5) = 4/36
P(6) = 5/36
P(7) = 6/36

A draw means that each gets the same score:
P(both score 2) = P(2) * P(2) = 1/36 * 1/36 = 1/1296
P(both score 3) = P(3) * P(3) = 2/36 * 2/36 = 4/1296
P(both score 4) = 3/6 * 3/36 = 9/1296
P(both score 5) = 4/36 * 4/36 = 16/1296
P(both score 6) = 5/36 * 5/36 = 25/36

P(both score 2 through 6) = (1+4+9+16+25)/1296 = 55/1296
Thus, P(both score 8 through 12) = 55/1295
P(both score 7) = 6/36 * 6/36 = 36/1296
Thus, P(D) = (55+55+36)/1296 = 146/1296

Thus P(J) = (1 - 146/1296)/2 = 575/1296.

The correct answer is C.

Last edited by GMATGuruNY on Sun Feb 26, 2012 5:08 am, edited 2 times in total.

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GMATGuruNY · Answer

yuliawati wrote:How to solve this problem quickly? I need more than 5 mins to get the answer. A long calculation....
Source: MGMAT

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?

(A) 15/36
(B) 175/432
(C) 575/1296
(D) 583/1296
(E) 1/2

OA later

3 possible outcomes:
P(J) = Probability that Jane wins
P(B) = Probability that Bill wins
P(D) = Probability in a draw (each gets the same score)

P(J) + P(B) + P(D) = 1.
Since Jane and Bill each have the same chance of winning, P(J) = P(B).
Thus:
P(J) + P(J) + P(D) = 1
2*P(J) + P(D) = 1
P(J) = [1- P(D)] / 2

So to determine P(J), we need to determine P(D).

The probabilities of getting the different sums display the following symmetry:

P(2) = P(12)
P(3) = P(11)
P(4) = P(10)
P(5) = P(9)
P(6) = P(8)

Let's determine the various probabilities to discern the pattern:
P(2) = P(1,1) = 1/6 * 1/6 = 1/36.
P(3) = P(1,2) + P(2,1) = 1/36 + 1/36 = 2/36
P(4) = P(1,3) + P(3,1) + P(2,2) = 3/36
Now we see the pattern:
P(5) = 4/36
P(6) = 5/36
P(7) = 6/36

A draw means that each gets the same score:
P(both score 2) = P(2) * P(2) = 1/36 * 1/36 = 1/1296
P(both score 3) = P(3) * P(3) = 2/36 * 2/36 = 4/1296
P(both score 4) = 3/6 * 3/36 = 9/1296
P(both score 5) = 4/36 * 4/36 = 16/1296
P(both score 6) = 5/36 * 5/36 = 25/36

P(both score 2 through 6) = (1+4+9+16+25)/1296 = 55/1296
Thus, P(both score 8 through 12) = 55/1295
P(both score 7) = 6/36 * 6/36 = 36/1296
Thus, P(D) = (55+55+36)/1296 = 146/1296

Thus P(J) = (1 - 146/1296)/2 = 575/1296.

The correct answer is C.

Last edited by GMATGuruNY on Sun Feb 26, 2012 5:08 am, edited 2 times in total.

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Rezinka · Answer

Great.. Thanks

yuliawati · Answer

Thank you, Rahul and Mitch! In dealing with the probability problem, my big mistake is careless calculation. That's very easy to make a mistake^^

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