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Math Expert V
Joined: 02 Sep 2009
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18 00:00

Difficulty:   85% (hard)

Question Stats: 47% (02:21) correct 53% (02:35) wrong based on 81 sessions

### HideShow timer Statistics Mary and Joe are to throw three dice each. The score is the sum of points on all three dice. If Mary scores 10 in her attempt what is the probability that Joe will outscore Mary in his?

A. $$\frac{24}{64}$$
B. $$\frac{32}{64}$$
C. $$\frac{36}{64}$$
D. $$\frac{40}{64}$$
E. $$\frac{42}{64}$$

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Math Expert V
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Official Solution:

Mary and Joe are to throw three dice each. The score is the sum of points on all three dice. If Mary scores 10 in her attempt what is the probability that Joe will outscore Mary in his?

A. $$\frac{24}{64}$$
B. $$\frac{32}{64}$$
C. $$\frac{36}{64}$$
D. $$\frac{40}{64}$$
E. $$\frac{42}{64}$$

To outscore Mary, Joe has to score in the range of 11-18. The probability to score 3 is the same as the probability to score 18 (1-1-1 combination against 6-6-6, if 1-1-1 is on the tops of the dice the 6-6-6 is on the bottoms). By the same logic, the probability to score $$x$$ is the same as the probability to score $$21 - x$$. Therefore, the probability to score in the range 11-18 equals the probability to score in the range of 3-10. As 3-18 covers all possible outcomes the probability to score in the range 11-18 is $$\frac{1}{2}$$ or $$\frac{32}{64}$$.

Alternative Explanation

Expected value of one die is $$\frac{1}{6}*(1+2+3+4+5+6)=3.5$$.

Expected value of three dice is $$3*3.5=10.5$$.

Mary scored 10 so the probability to get the sum more than 10 (11, 12, 13, ..., 18), or more than the average, is the same as to get the sum less than average (10, 9, 8, ..., 3) = 1/2 = 32/64.

That's because the probability distribution is symmetrical for this case:

The probability of getting the sum of 3 (min possible sum) = the probability of getting the sum of 18 (max possible sum);

The probability of getting the sum of 4 = the probability of getting the sum of 17;

The probability of getting the sum of 5 = the probability of getting the sum of 16;

...

The probability of getting the sum of 10 = the probability of getting the sum of 11;

Thus the probability of getting the sum from 3 to 10 = the probability of getting the sum from 11 to 18 = 1/2.

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Bunuel can you please how the number 21 is derived in the original explanation?

Also, here we know that the average is 10.5 & not 10 so how are we deducing that half the times sum will be less than 10 ( & not 10.5 ) & half the time ( sum will be more than 10.5) .

I can understand since number showing up on the dice being integers we will never get 10.5 ( it will be either 10 or 11 but never 10.5). Nonetheless, technically speaking do we need to say that half the times sum will be greater than average & half the time it will be lesser.
Math Expert V
Joined: 02 Sep 2009
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ankushbagwale wrote:
Bunuel can you please how the number 21 is derived in the original explanation?

Also, here we know that the average is 10.5 & not 10 so how are we deducing that half the times sum will be less than 10 ( & not 10.5 ) & half the time ( sum will be more than 10.5) .

I can understand since number showing up on the dice being integers we will never get 10.5 ( it will be either 10 or 11 but never 10.5). Nonetheless, technically speaking do we need to say that half the times sum will be greater than average & half the time it will be lesser.

The probability of scoring 1 + 1 + 1 = 3 is the same as the probability of scoring 6 + 6 + 6 = 18 = 21 - 3.
The probability of scoring 2 + 1 + 1 = 4 is the same as the probability of scoring 6 + 6 + 5 = 17 = 21 - 4.
...
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Bunuel wrote:
Official Solution:

Mary and Joe are to throw three dice each. The score is the sum of points on all three dice. If Mary scores 10 in her attempt what is the probability that Joe will outscore Mary in his?

A. $$\frac{24}{64}$$
B. $$\frac{32}{64}$$
C. $$\frac{36}{64}$$
D. $$\frac{40}{64}$$
E. $$\frac{42}{64}$$

To outscore Mary, Joe has to score in the range of 11-18. The probability to score 3 is the same as the probability to score 18 (1-1-1 combination against 6-6-6, if 1-1-1 is on the tops of the dice the 6-6-6 is on the bottoms). By the same logic, the probability to score $$x$$ is the same as the probability to score $$21 - x$$. Therefore, the probability to score in the range 11-18 equals the probability to score in the range of 3-10. As 3-18 covers all possible outcomes the probability to score in the range 11-18 is $$\frac{1}{2}$$ or $$\frac{32}{64}$$.

Alternative Explanation

Expected value of one die is $$\frac{1}{6}*(1+2+3+4+5+6)=3.5$$.

Expected value of three dice is $$3*3.5=10.5$$.

Mary scored 10 so the probability to get the sum more than 10 (11, 12, 13, ..., 18), or more than the average, is the same as to get the sum less than average (10, 9, 8, ..., 3) = 1/2 = 32/64.

That's because the probability distribution is symmetrical for this case:

The probability of getting the sum of 3 (min possible sum) = the probability of getting the sum of 18 (max possible sum);

The probability of getting the sum of 4 = the probability of getting the sum of 17;

The probability of getting the sum of 5 = the probability of getting the sum of 16;

...

The probability of getting the sum of 10 = the probability of getting the sum of 11;

Thus the probability of getting the sum from 3 to 10 = the probability of getting the sum from 11 to 18 = 1/2.

Can you please list out 32 scenarios? I tried to list out but found only 30 as below:
146,156,166
236,246,256,266
326,336,346,356,366
416,426,436,446,456,466
516,526,536,546,556,566
616,626,636,646,656,666
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Hi - Can someone explain a bit more about the "expected value"? What does this math mean?
Math Expert V
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glochou wrote:
Hi - Can someone explain a bit more about the "expected value"? What does this math mean?

Hi,

when you throw a dice, it can give you any value 1 to 6....
since any of the six faces can be facing up, the probability of each was 1/6....
so total value taking this probability is$$\frac{1}{6} *1 + \frac{1}{6} *2+ \frac{1}{6} *3 + \frac{1}{6} *4+ \frac{1}{6} *5 + \frac{1}{6} *6$$ = $$\frac{1}{6}(1+2+3+4+5+6)$$ = $$\frac{21}{6}$$ = 3.5
this is the expected value on each trow
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I think this is a high-quality question and I agree with explanation. I solved like this :

Possible score values for M : 3,4,5,6,7,8,9,10
Possible score values for J : 11,12,13,14,15,16,17,18 ---> 8 favorable events

P(Joe will Outscore M ) = number of favorable conditions / Total number of conditions.

= 8/16 = 1/2 => 32/64

Considering my approach is right, I was wondering why this question falls under 95% hard or 700 + level question. It triggered to me like a 500 level question.
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dharan wrote:
I think this is a high-quality question and I agree with explanation. I solved like this :

Possible score values for M : 3,4,5,6,7,8,9,10
Possible score values for J : 11,12,13,14,15,16,17,18 ---> 8 favorable events

P(Joe will Outscore M ) = number of favorable conditions / Total number of conditions.

= 8/16 = 1/2 => 32/64

Considering my approach is right, I was wondering why this question falls under 95% hard or 700 + level question. It triggered to me like a 500 level question.

The logic is not correct. Check here: mary-and-joe-are-to-throw-three-dice-each-the-score-is-the-86407.html#p788093
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I think this is a high-quality question and I agree with explanation.
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Hi, Please help me understand the explanation. I did not understand the relevance of Expected Value and how does it help us in solving the question. ?
Also, what if Mary scored 13 and we need to find Probability of having score more than 13, how would we solve then? Pls help. Thanks.
Math Expert V
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Saurav Arora wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Hi, Please help me understand the explanation. I did not understand the relevance of Expected Value and how does it help us in solving the question. ?
Also, what if Mary scored 13 and we need to find Probability of having score more than 13, how would we solve then? Pls help. Thanks.

Check the discussion here: mary-and-joe-are-to-throw-three-dice-each-the-score-is-the-86407.html
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Possible score values for M : 3,4,5,6,7,8,9,10
Possible score values for J : 11,12,13,14,15,16,17,18 ---> 8 favorable events
Is this a suitable way of attempting the Q...Probability of individual events are different right.Because it is evenly distributed this method can be applied...For a general Q can this be used??
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VyshakhR1995 wrote:
Possible score values for M : 3,4,5,6,7,8,9,10
Possible score values for J : 11,12,13,14,15,16,17,18 ---> 8 favorable events
Is this a suitable way of attempting the Q...Probability of individual events are different right.Because it is evenly distributed this method can be applied...For a general Q can this be used??

Check here: http://gmatclub.com/forum/mary-and-joe- ... ml#p788093
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please confirm if following method is correct:

to get a sum more than 10, two dices shall have values 4 or more and the third shall have 3 or more
1st dice: 4 or more
probab = 3/6

2nd dice: 4 or more
probab = 3/6

3rd dice: 3 ore more
probab = 4/6

now final probability =
$$\frac{3}{6}*\frac{3}{6}*\frac{4}{6}*\frac{3*2}{2}=\frac{1}{2}$$

multiplied by $$\frac{3*2}{2}$$ as there are 3*2 ways in the 3 dice and 2 are similar so divide by 2 so final will be $$\frac{3*2}{2}$$
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asthagupta wrote:
please confirm if following method is correct:

to get a sum more than 10, two dices shall have values 4 or more and the third shall have 3 or more
1st dice: 4 or more
probab = 3/6

2nd dice: 4 or more
probab = 3/6

3rd dice: 3 ore more
probab = 4/6

now final probability =
$$\frac{3}{6}*\frac{3}{6}*\frac{4}{6}*\frac{3*2}{2}=\frac{1}{2}$$

multiplied by $$\frac{3*2}{2}$$ as there are 3*2 ways in the 3 dice and 2 are similar so divide by 2 so final will be $$\frac{3*2}{2}$$

Are you sure about coloured portion..
what about two as 3 and 3rd as 5 or 6..
3,3,5... total 11
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I think this is a high-quality question and I agree with explanation.
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Great question Could you please tell the source ??
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spetznaz wrote:
Great question Could you please tell the source ??

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Please do let me know whether this method of mine is correct?
@bunnel chetan2u
Joe has score 11or more to outscore Mary.
1+4+6=11( On first dice 1, then on second dice 4 and then on the third dice 6)
On Dice 1, there are 6 options( 1,2,3,4,5,6)
On Dice 2, there are 3 options(4,5,6)
On Dice 3, only 1 option(6)
The minimum sum will be 11 which is the minimum required condition.
for sum to be 11 or more, simultaneous action is required.
Hence,
6C1*3C1*1C1=18
these numbers on the dices can be arranged in !3=6 ways
so number of outcome=18*6
total outcome=6*6*6
Required Probability=18*6/(6*6*6*)
=1/2 M08-21   [#permalink] 02 Nov 2018, 06:40

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# M08-21

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