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# A conjuror will roll one red, six-sided die in his right hand and two

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Math Expert
Joined: 02 Sep 2009
Posts: 42649

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A conjuror will roll one red, six-sided die in his right hand and two [#permalink]

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20 Jul 2017, 21:01
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Difficulty:

75% (hard)

Question Stats:

50% (02:06) correct 50% (01:39) wrong based on 58 sessions

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A conjuror will roll one red, six-sided die in his right hand and two blue, six-sided dice. What is the probability that the number on the red die will be greater than the sum of the two blue dice?

(A) 5/54
(B) 5/108
(C) 11/216
(D) 7/36
(E) 5/18
[Reveal] Spoiler: OA

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Kudos [?]: 135955 [0], given: 12717

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A conjuror will roll one red, six-sided die in his right hand and two [#permalink]

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21 Jul 2017, 02:40
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In total, there are ($$6^3 = 216$$) combinations.

The minimum sum on the 2 blue die is 2(1+1).
Since we have been asked the probability of red die's sum
to be greater than the blue die's, the minimum number we can have on red die is 3. (1 combination)

When we get a 4 on the red die, there are 3 combinations for the blue die{(1,2),(2,1),(1,1)}

When we get a 5 on the red die, there are 6 combinations for the blue die{(1,2),(2,1),(1,1),(2,2),(1,3),(3,1)}

When we get a 6 on the red die, there are 10 combinations for the blue die{(1,2),(2,1),(1,1),(2,2),(1,3),(3,1),(2,3),(3,2),(1,4),(4,1)}

The probability that the number on the red die will be greater than the sum of the two blue dice = $$\frac{1+3+6+10}{216} = \frac{20}{216} = \frac{5}{54}$$(Option A)
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Kudos [?]: 754 [0], given: 20

EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
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Location: United States (CA)
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GRE 1: 340 Q170 V170
Re: A conjuror will roll one red, six-sided die in his right hand and two [#permalink]

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22 Nov 2017, 19:22
1
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Expert's post
Hi All,

This prompt asks for the probability that the number on one 6-sided die will be greater than the SUM of the numbers on two 6-sided dice. This question requires us to consider multiple possible situations.

To start, we only have to consider a few possible sums for the 'pair' of dice: 2, 3, 4 and 5 (in all other situations, the sum of the two dice CANNOT be less than the total on the one die).

Probability of rolling a total of 2 on two dice (1 and 1) and higher than 2 on one die = (1/6)(1/6)(4/6) = 4/216
Probability of rolling a total of 3 on two dice (1 and 2 in some order) and higher than 3 on one die = (2/6)(1/6)(3/6) = 6/216
Probability of rolling a total of 4 on two dice (1 and 3 or 2 and 2, in some order) and higher than 4 on one die = (3/6)(1/6)(2/6) = 6/216
Probability of rolling a total of 5 on two dice (1 and 4 or 2 and 3, in some order) and higher than 5 on one die = (4/6)(1/6)(1/6) = 4/216

Total = 4/216 + 6/216 + 6/126 + 4/216 = 20/216 = 5/54

[Reveal] Spoiler:
A

GMAT assassins aren't born, they're made,
Rich
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Kudos [?]: 3700 [1], given: 173

Senior Manager
Joined: 09 Feb 2015
Posts: 376

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Location: India
Concentration: Social Entrepreneurship, General Management
GMAT 1: 690 Q49 V34
GMAT 2: 720 Q49 V39
GPA: 2.8
Re: A conjuror will roll one red, six-sided die in his right hand and two [#permalink]

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26 Nov 2017, 01:09
EMPOWERgmatRichC wrote:
Hi All,

This prompt asks for the probability that the number on one 6-sided die will be greater than the SUM of the numbers on two 6-sided dice. This question requires us to consider multiple possible situations.

To start, we only have to consider a few possible sums for the 'pair' of dice: 2, 3, 4 and 5 (in all other situations, the sum of the two dice CANNOT be less than the total on the one die).

Probability of rolling a total of 2 on two dice (1 and 1) and higher than 2 on one die = (1/6)(1/6)(4/6) = 4/216
Probability of rolling a total of 3 on two dice (1 and 2 in some order) and higher than 3 on one die = (2/6)(1/6)(3/6) = 6/216
Probability of rolling a total of 4 on two dice (1 and 3 or 2 and 2, in some order) and higher than 4 on one die = (3/6)(1/6)(2/6) = 6/216
Probability of rolling a total of 5 on two dice (1 and 4 or 2 and 3, in some order) and higher than 5 on one die = (4/6)(1/6)(1/6) = 4/216

Total = 4/216 + 6/216 + 6/126 + 4/216 = 20/216 = 5/54

[Reveal] Spoiler:
A

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

if it isnt mentioned that the dice are different do we assume that they are in gmat?

Kudos [?]: 6 [0], given: 199

Re: A conjuror will roll one red, six-sided die in his right hand and two   [#permalink] 26 Nov 2017, 01:09
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