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

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Posts: 52231
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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3
00:00

Difficulty:

85% (hard)

Question Stats:

53% (02:54) correct 47% (02:44) wrong based on 99 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

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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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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
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

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# Rich Cohen

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** Senior Manager Joined: 09 Feb 2015 Posts: 359 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] ### Show Tags 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 Final Answer: 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? Senior Manager Joined: 08 Jun 2015 Posts: 432 Location: India GMAT 1: 640 Q48 V29 GMAT 2: 700 Q48 V38 GPA: 3.33 Re: A conjuror will roll one red, six-sided die in his right hand and two [#permalink] ### Show Tags 23 Apr 2018, 06:47 goforgmat wrote: 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 Final Answer: 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? Exactly my doubt ... EMPOWERgmatRichC pl clarify ... _________________ " The few , the fearless " EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 13341 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: A conjuror will roll one red, six-sided die in his right hand and two [#permalink] ### Show Tags 23 Apr 2018, 09:45 Hi goforgmat & spetznaz, I'm not sure what you mean when you ask about whether the two dice are 'different' or not. When we're talking about the SUM of two dice rolls, it does not matter whether you roll two different dice or the same one die twice (the possible outcomes would be the same). In addition, since each die roll is independent of the other, it's possible that the two individual rolls could be the same (for example, 1 and 1). If there are specific aspects/examples to this question that you would like to discuss in more detail, then just let me know. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: A conjuror will roll one red, six-sided die in his right hand and two &nbs [#permalink] 23 Apr 2018, 09:45
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