GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 14 Nov 2018, 13:56

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel
Events & Promotions in November
PrevNext
SuMoTuWeThFrSa
28293031123
45678910
11121314151617
18192021222324
2526272829301
Open Detailed Calendar
  • $450 Tuition Credit & Official CAT Packs FREE

     November 15, 2018

     November 15, 2018

     10:00 PM MST

     11:00 PM MST

    EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth $100 with the 3 Month Pack ($299)
  • Free GMAT Strategy Webinar

     November 17, 2018

     November 17, 2018

     07:00 AM PST

     09:00 AM PST

    Nov. 17, 7 AM PST. Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.

Romeo and Juliet play a dice game in which the two participants take..

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Director
Director
User avatar
D
Joined: 11 Feb 2015
Posts: 562
Premium Member CAT Tests
Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 27 Oct 2018, 09:29
3
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

56% (01:40) correct 44% (01:30) wrong based on 62 sessions

HideShow timer Statistics

Romeo and Juliet play a dice game in which the two participants take turns rolling a single fair six-sided die. The first player to roll a 6 wins. If Romeo rolls first, what is the probability that Juliet will win?

A) \(\frac{1}{4}\)
B) \(\frac{1}{3}\)
C) \(\frac{4}{9}\)
D) \(\frac{5}{11}\)
E) \(\frac{1}{2}\)

_________________

"Please hit :thumbup: +1 Kudos if you like this post" :student_man:

_________________
Manish :geek:

"Only I can change my life. No one can do it for me"

Director
Director
User avatar
D
Joined: 11 Feb 2015
Posts: 562
Premium Member CAT Tests
Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 27 Oct 2018, 11:29
1
Official Answer from Veritas prep:

The key to this difficult probability question is focusing not on Romeo’s win probability, not on Juliet’s win probability, but instead on the relationship between the two. Consider Juliet’s chances as the sum of two cases: Either, with probability \(\frac{1}{6}\) , Romeo gets a 6 on the first roll – Juliet loses; or, with probability \(\frac{5}{6}\) , Romeo does not get a 6 on the first roll – Juliet becomes Romeo. Of course, Juliet does not literally become Romeo in the latter case, but the point is that Romeo’s and Juliet’s situations and probabilities are then reversed: Juliet gets to roll with a chance to win on a 6 or else pass the turn to her opponent, so her probability at that moment is exactly whatever Romeo’s win probability was at the beginning of the game.

This may sound messy, but algebraically it’s quite neat:


\(PJ=\frac{1}{6}∗0+\frac{5}{6}∗PR=\frac{5}{6}PR\)


Of course, someone will win the game eventually, which means that the two probabilities are complementary, and we can also write


\(PJ+PR=1\)


Now we have a system of two equations and two variables, and substitution will clean it up nicely:


\(\frac{5}{6}PR+PR=1\)


\(\frac{11}{6}\)PR=1


PR=\(\frac{6}{11}\)
and
PJ=\(\frac{5}{6}∗\frac{6}{11}=\frac{5}{11}\)


This is the correct answer, D.
_________________

"Please hit :thumbup: +1 Kudos if you like this post" :student_man:

_________________
Manish :geek:

"Only I can change my life. No one can do it for me"

Director
Director
User avatar
D
Joined: 11 Feb 2015
Posts: 562
Premium Member CAT Tests
Re: Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 27 Oct 2018, 11:50
2
CAMANISHPARMAR wrote:
Romeo and Juliet play a dice game in which the two participants take turns rolling a single fair six-sided die. The first player to roll a 6 wins. If Romeo rolls first, what is the probability that Juliet will win?

A) \(\frac{1}{4}\)
B) \(\frac{1}{3}\)
C) \(\frac{4}{9}\)
D) \(\frac{5}{11}\)
E) \(\frac{1}{2}\)


(Lets consider the SIMPLE CASE first) the probability of Juliet winning the game considering Romeo looses in first turn and Juliet wins in second turn was \(\frac{5}{6}*\frac{1}{6}\) = \(\frac{5}{36}\)

But what if Juliet does not win in the her first turn but wins in her second turn. Then we have a scenerio where Romeo looses, Juliet also looses, again Romeo looses & finally Juliet wins which translates into \(\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}\)

The total probability that juliet wins in either of her first two chances will be sum of the individual probabilities of she either winning in her first or second chance.

= \(\frac{5}{6}*\frac{1}{6}\)+ \(\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}\)

Similarly:-

The total probability that juliet wins in either of the first THREE chances is

= \(\frac{5}{6}*\frac{1}{6}\) + \(\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}\) + \(\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{5}{6} *\frac{5}{6}*\frac{1}{6}\)

We can continue like this towards infinity & if we take out \(\frac{5}{6}*\frac{1}{6}\) from this series, we realise that this is a geometric progression with common ratio, r = \(\frac{5}{6}*\frac{5}{6}\)

If we add the probabilities of all individual possible events... we have....
\(\frac{5}{6}*\frac{1}{6}\) + \(\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}\) + \(\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{5}{6} *\frac{5}{6}*\frac{1}{6}\) + ..... so on

Note the formula for a sum of GP is:-
Attachment:
GP sum.jpg
GP sum.jpg [ 2.28 KiB | Viewed 661 times ]


Since common ratio of this GP is less than one then \(r^n\) (where n is infinite) means \(r^n\) will become very minute and we can ignore it.
Attachment:
Expo.jpg
Expo.jpg [ 19.49 KiB | Viewed 658 times ]


Therefore substituting the values of a = \(\frac{5}{6}*\frac{1}{6}\) and \(r^n\) = 0 and r = \(\frac{5}{6}*\frac{5}{6}\) we get:-

\(\frac{5}{6}*\frac{1}{6}\)*\(\frac{36}{11}\) = \(\frac{5}{11}\) (Ans)

Hence option D is the correct option.
_________________

"Please hit :thumbup: +1 Kudos if you like this post" :student_man:

_________________
Manish :geek:

"Only I can change my life. No one can do it for me"

Intern
Intern
avatar
B
Joined: 17 Jun 2017
Posts: 26
Re: Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 04 Nov 2018, 00:38
CAMANISHPARMAR wrote:
Romeo and Juliet play a dice game in which the two participants take turns rolling a single fair six-sided die. The first player to roll a 6 wins. If Romeo rolls first, what is the probability that Juliet will win?

A) \(\frac{1}{4}\)
B) \(\frac{1}{3}\)
C) \(\frac{4}{9}\)
D) \(\frac{5}{11}\)
E) \(\frac{1}{2}\)


This can't be a two-minute question as per the solutions provided!
Manager
Manager
avatar
G
Joined: 27 Dec 2016
Posts: 241
Re: Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 08 Nov 2018, 18:59
Bunuel chetan2u
Hi experts, I was wondering if there was an easier way to solve this problem? The solutions given above will take more than 2 mins to solve.
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7029
Re: Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 09 Nov 2018, 06:29
csaluja wrote:
Bunuel chetan2u
Hi experts, I was wondering if there was an easier way to solve this problem? The solutions given above will take more than 2 mins to solve.



Hi..

I) Logically if you look at the question, you can eliminate 3 choices..
Both have fair chances, so should have almost equal chances. A and B choice give very less probability to Juliet, so eliminate them
But only difference is that Romeo gets the first try, so his probability should be more and both cannot be equal. hence eliminate E
C and D are close but your probability becomes 50% to choose the correct choice.

II) Geometric Progression
It is a GP question and can be solved in a minute if you know the formula for an infinity....
J can fin in his first chance only if R does not win on first chance.. so R can pick any remaining 5 so \(\frac{5}{6}\), then J has to pick 6 so \(\frac{1}{6}\)..P = \(\frac{5}{6}*\frac{1}{6}\)
J can win in his next throw, if the first three throws have had any of other 5 numbers thus \(\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}\)
So P = \(\frac{5}{6}*\frac{1}{6}\)+\(\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}\)+.....
so r = \(\frac{5}{6}^2\)
formula is \(\frac{a}{1-r}\)=\((\frac{5}{6}*\frac{1}{6})/(1-\frac{5}{6}^2)=(\frac{5}{36})/\frac{11}{36}=\frac{5}{11}\)
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

Manager
Manager
avatar
B
Joined: 18 Aug 2017
Posts: 184
Concentration: Healthcare, Marketing
CAT Tests
Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 09 Nov 2018, 23:03
Romeo and Juliet play a dice game in which the two participants take turns rolling a single fair six-sided die. The first player to roll a 6 wins. If Romeo rolls first, what is the probability that Juliet will win?

A) 1/4
B) 1/3
C) 4/9
D) 5/11
E) 1/2

GMATinsight: Sir is there any other way to solve this probability question apart from using GP expression which other experts have also used?
CEO
CEO
User avatar
P
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2698
Location: India
GMAT: INSIGHT
WE: Education (Education)
Reviews Badge
Re: Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 09 Nov 2018, 23:19
1
Archit3110 wrote:
Romeo and Juliet play a dice game in which the two participants take turns rolling a single fair six-sided die. The first player to roll a 6 wins. If Romeo rolls first, what is the probability that Juliet will win?

A) 1414
B) 1313
C) 4949
D) 511511
E) 12

GMATinsight: Sir is there any other way to solve this probability question apart from using GP expression which other experts have also used?


No, I am unable to think any alternative method to solve this other than coming down to the following calculation

For juliet to win

Case 1: Romeo gets a number other than 6 and juliet in first throw gets 6
Probability \(= (5/6)*(1/6)\)

Case 2: Romeo gets a number other than 6 on two successive throws and juliet in first throw gets other than 6 and in second throw gets a 6
Probability \(= (5/6)*(5/6)*(5/6)*(1/6) = (5/6)^3*(1/6)\)

Case 3: Romeo gets a number other than 6 on three successive throws and juliet in first two throw gets other than 6 and in third throw gets a 6
Probability \(= (5/6)*(5/6)*(5/6)*(5/6)*(5/6)*(1/6) = (5/6)^5*(1/6)\)

and so on...

Total Probability \(= (5/6)*(1/6) + (5/6)^3*(1/6) + (5/6)^5*(1/6) + --- = (1/6)*[(5/6)+(125/216)+---] =\)

which eventually requires sum of an infinite GP which may conveniently be called beyond GMAT scope.

Archit3110
_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

Veritas Prep GMAT Instructor
User avatar
P
Joined: 16 Oct 2010
Posts: 8531
Location: Pune, India
Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 10 Nov 2018, 05:21
1
1
CAMANISHPARMAR wrote:
Romeo and Juliet play a dice game in which the two participants take turns rolling a single fair six-sided die. The first player to roll a 6 wins. If Romeo rolls first, what is the probability that Juliet will win?

A) \(\frac{1}{4}\)
B) \(\frac{1}{3}\)
C) \(\frac{4}{9}\)
D) \(\frac{5}{11}\)
E) \(\frac{1}{2}\)


You don't need to use the GP formula if you don't want to. Here is how:

\(P(Romeo) = \frac{1}{6} + (\frac{5}{6}*\frac{5}{6}*\frac{1}{6}) + (\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}) + ...\) ... (I)

\(P(Juliet) = (\frac{5}{6}*\frac{1}{6}) + (\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}) + ...\) ... (II)

Substituting (II) into (I), we get

\(P(Romeo) = \frac{1}{6} + (\frac{5}{6})*P(Juliet)\)

But, \(P(Romeo) + P(Juliet) = 1\)
So,
\(1 - P(Juliet) = \frac{1}{6} + (\frac{5}{6})*P(Juliet)\)

\(P(Juliet) = \frac{5}{11}\)

Answer (D)
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Director
Director
User avatar
D
Joined: 11 Feb 2015
Posts: 562
Premium Member CAT Tests
Re: Romeo and Juliet play a dice game in which the two participants take..  [#permalink]

Show Tags

New post 10 Nov 2018, 05:25
VeritasKarishma wrote:
CAMANISHPARMAR wrote:
Romeo and Juliet play a dice game in which the two participants take turns rolling a single fair six-sided die. The first player to roll a 6 wins. If Romeo rolls first, what is the probability that Juliet will win?

A) \(\frac{1}{4}\)
B) \(\frac{1}{3}\)
C) \(\frac{4}{9}\)
D) \(\frac{5}{11}\)
E) \(\frac{1}{2}\)


You don't need to use the GP formula if you don't want to. Here is how:

\(P(Romeo) = \frac{1}{6} + (\frac{5}{6}*\frac{5}{6}*\frac{1}{6}) + (\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}) + ...\) ... (I)

\(P(Juliet) = (\frac{5}{6}*\frac{1}{6}) + (\frac{5}{6}*\frac{5}{6}*\frac{5}{6}*\frac{1}{6}) + ...\) ... (II)

Substituting (II) into (I), we get

\(P(Romeo) = \frac{1}{6} + (\frac{5}{6})*P(Juliet)\)

But, \(P(Romeo) + P(Juliet) = 1\)
So,
\(1 - P(Juliet) = \frac{1}{6} + (\frac{5}{6})*P(Juliet)\)

\(P(Juliet) = \frac{5}{11}\)

Answer (D)


Your thinking is always supercool :) ...... you always think out of the box!! Needless to say that this post deserves a kudos :)
_________________

"Please hit :thumbup: +1 Kudos if you like this post" :student_man:

_________________
Manish :geek:

"Only I can change my life. No one can do it for me"

GMAT Club Bot
Re: Romeo and Juliet play a dice game in which the two participants take.. &nbs [#permalink] 10 Nov 2018, 05:25
Display posts from previous: Sort by

Romeo and Juliet play a dice game in which the two participants take..

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.