December 16, 2018 December 16, 2018 03:00 PM EST 04:00 PM EST Strategies and techniques for approaching featured GMAT topics December 16, 2018 December 16, 2018 07:00 AM PST 09:00 AM PST Get personalized insights on how to achieve your Target Quant Score.
Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 19 Apr 2011
Posts: 199
Schools: Booth,NUS,St.Gallon

A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
23 Sep 2012, 02:34
Question Stats:
22% (02:32) correct 78% (02:31) wrong based on 195 sessions
HideShow timer Statistics
A fair coin is tossed 6 times. What is the probability of getting no any two heads on consecutive tosses? A. 21/64 B. 42/64 C. 19/64 D. 19/42 E. 31/64
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
+1 if you like my explanation .Thanks




Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4489

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
24 Sep 2012, 21:34
saikarthikreddy wrote: A fair coin is tossed 6 times. What is the probability of getting no any two heads on consecutive tosses? a.21/64 b.42/64 c.19/64 d.19/42 e.31/64 Hi, there. I'm happy to help with this. First of all, question is considerably harder and more paininthetush than what you will see on the GMAT. I don't know the source, but this seems to come from some overachieving source that wants to give students questions much harder than the test. So this is a probability question that is best solved with counting. You may find this blog germane: http://magoosh.com/gmat/2012/gmatquanthowtocount/Probability = (# of desired cases)/(total # of possible cases) The denominator is very easy  two possibilities for each toss, six tosses, so 2^6 = 64. That's the denominator. For the numerator, we have to sort through cases: Case One: six tailsFor this case, obviously you can't have two heads in a row. There's only one way this can happen: TTTTTT ONECase Two: five tails, one headAgain, it's impossible to have two heads in a row, because there's only one. There are six ways this could happen  the H could occupy any of the six positions (HTTTTT, THTTTT, TTHTTT, TTTHTT, TTTTHT, and TTTTTH) SIXCase Three: four tails, two headsThis is the tricky case. There are 6C2 = 15 places that the two H's could land, but five of those (HHTTTT, THHTTT, TTHHTT, TTTHHT, and TTTTHH) involve the pair of H's together, which is forbidden. Excluding those five forbidden cases, we are left with 15  5 = 10 possibilities here. TENCase Four: three tails, three headsNow, things are starting to get crowded. We have to space our three H's out, with T's between them, so none of the H's touch. That leaves only two possibilities: HTHTHT and THTHTH. That's it: any other configuration would have two H's next to each other, which is forbidden. TWOAdd those up: 1 + 6 + 10 + 2 = 19. That's our numerator. Probability = 19/64 Answer = CDoes all that make sense? Please let me know if you have any further questions. Mike
_________________
Mike McGarry Magoosh Test Prep
Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)




Intern
Joined: 20 Sep 2012
Posts: 49
Concentration: Finance, Entrepreneurship
GPA: 3.37
WE: Analyst (Consulting)

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
Updated on: 24 Sep 2012, 22:02
A There are 64 possible outcomes.  If no head => 1 outcome  If we can only toss head once then there are 6 desired outcomes (eg: HTTTTT, THTTTT....)  If we can toss head twice then there are \(\frac{6!}{2!*4!}5 = 10\) desired outcomes.  If we can toss head three times then there are 4 outcomes THTHTH, HTHTHT, HTTHTH, HTHTTH  4, 5, 6 times > no outcome => Probability = \(\frac{21}{64}\)
Originally posted by monsama on 24 Sep 2012, 21:57.
Last edited by monsama on 24 Sep 2012, 22:02, edited 3 times in total.



Intern
Joined: 20 Sep 2012
Posts: 49
Concentration: Finance, Entrepreneurship
GPA: 3.37
WE: Analyst (Consulting)

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
24 Sep 2012, 21:59
mikemcgarry wrote: saikarthikreddy wrote: A fair coin is tossed 6 times. What is the probability of getting no any two heads on consecutive tosses? a.21/64 b.42/64 c.19/64 d.19/42 e.31/64 Case Four: three tails, three headsNow, things are starting to get crowded. We have to space our three H's out, with T's between them, so none of the H's touch. That leaves only two possibilities: HTHTHT and THTHTH. That's it: any other configuration would have two H's next to each other, which is forbidden. TWO@mikemcgarry: in case 4, what about HTTHTH and HTHTTH => there are 4 possibilities :D



Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4489

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
25 Sep 2012, 08:19
MonSama wrote: @mikemcgarry: in case 4, what about HTTHTH and HTHTTH => there are 4 possibilities :D Very good! I stand corrected. The answer must be (A), as MonSama suggests. Mike
_________________
Mike McGarry Magoosh Test Prep
Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)



Manager
Joined: 19 Apr 2011
Posts: 199
Schools: Booth,NUS,St.Gallon

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
25 Sep 2012, 23:09
Thanks for the comprehensive solution to both Mike and Monsama .. Kudos for both of you !!
_________________
+1 if you like my explanation .Thanks



Senior Manager
Joined: 13 Aug 2012
Posts: 429
Concentration: Marketing, Finance
GPA: 3.23

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
26 Sep 2012, 00:24
All possibilities with no 2 H touching each other.
Attachments
solution mixture.jpg [ 24.64 KiB  Viewed 40379 times ]
_________________
Impossible is nothing to God.



Manager
Joined: 22 Feb 2009
Posts: 171

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
31 Jul 2014, 14:59
saikarthikreddy wrote: A fair coin is tossed 6 times. What is the probability of getting no any two heads on consecutive tosses? a.21/64 b.42/64 c.19/64 d.19/42 e.31/64 I found a similar question in the link below afaircoinistossed5timeswhatistheprobabilityof99478.html
_________________
......................................................................... +1 Kudos please, if you like my post



Manager
Joined: 01 Apr 2015
Posts: 50

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
20 Aug 2015, 05:39
Bunuel, is there any other way than counting method to solve this question ? Thanks.



Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4489

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
20 Aug 2015, 09:22
Swaroopdev wrote: Bunuel, is there any other way than counting method to solve this question? Thanks. Dear SwaroopdevI'm happy to respond. My friend, a couple things to keep in mind. When Bunuel and the other experts show a solution to a problem, a relatively long and complicated solution, it's not as if we are making it complicated just for our own amusement. If there were a quick, easy, formulaic way to approach the problem, of course we would show that. In general, I would say that you can trust Bunuel and the other experts to show you the easiest, the most straightforward, and most efficient solution to any problem. Saying, " This solution looks hard. I don't like it. Can you show me an easier way?" is not the path that leads to excellence. The path that leads to excellence is all about challenging oneself to dive into what is most difficult and confusing. Assume that the path that Bunuel shows you is the optimal solution, and do your best to understand every last detail of it. Also, there's something important thing to keep in mind about Counting and about Probability. Other branches of math, such as algebra, tend to be more formulaic and recipe based. If I give you a simple algebraic equation and ask you to solve for x, there's an easy recipe to follow. Counting and Probability are not primarily formulabased or recipebased. Yes, there are a few formulas, but it is far from straightforward to know exactly when they can or can't be applied. Many Counting & Probability problems are about seeing the problem correct, interpreting the given information in a way that allows you to dissect the problem. Many students get into a "what should I do?" mode in problem solving, and with both Counting & Probability, it's important at the beginning of a problem to be not in the "what should I do?" mode, but in the "how do I look at this?" mode. When you are looking at the problem in the right way, what to do become obvious. Here's a blog with a few challenging counting problems and more on the mindset you need to cultivate to be successful with these problems. http://magoosh.com/gmat/2013/difficult ... problems/Does all this make sense? Mike
_________________
Mike McGarry Magoosh Test Prep
Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)



CEO
Joined: 20 Mar 2014
Posts: 2633
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
20 Aug 2015, 09:28
mikemcgarry wrote: Swaroopdev wrote: Bunuel, is there any other way than counting method to solve this question? Thanks. Dear SwaroopdevI'm happy to respond. My friend, a couple things to keep in mind. When Bunuel and the other experts show a solution to a problem, a relatively long and complicated solution, it's not as if we are making it complicated just for our own amusement. If there were a quick, easy, formulaic way to approach the problem, of course we would show that. In general, I would say that you can trust Bunuel and the other experts to show you the easiest, the most straightforward, and most efficient solution to any problem. Saying, " This solution looks hard. I don't like it. Can you show me an easier way?" is not the path that leads to excellence. The path that leads to excellence is all about challenging oneself to dive into what is most difficult and confusing. Assume that the path that Bunuel shows you is the optimal solution, and do your best to understand every last detail of it. Also, there's something important thing to keep in mind about Counting and about Probability. Other branches of math, such as algebra, tend to be more formulaic and recipe based. If I give you a simple algebraic equation and ask you to solve for x, there's an easy recipe to follow. Counting and Probability are not primarily formulabased or recipebased. Yes, there are a few formulas, but it is far from straightforward to know exactly when they can or can't be applied. Many Counting & Probability problems are about seeing the problem correct, interpreting the given information in a way that allows you to dissect the problem. Many students get into a "what should I do?" mode in problem solving, and with both Counting & Probability, it's important at the beginning of a problem to be not in the "what should I do?" mode, but in the "how do I look at this?" mode. When you are looking at the problem in the right way, what to do become obvious. Here's a blog with a few challenging counting problems and more on the mindset you need to cultivate to be successful with these problems. http://magoosh.com/gmat/2013/difficult ... problems/Does all this make sense? Mike Excellent post, mikemcgarryA big fan of your posts especially the one on different levels of "understanding". Keep them coming.



Manager
Joined: 01 Apr 2015
Posts: 50

A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
20 Aug 2015, 10:36
Hi mikemcgarry, Thank you for your response. First off, i have read and learnt so much from all your posts both in verbal and quant, i really appreciate your time and effort for this too. Also, i didn't mention this in my previous post but i didn't ask Bunuel for alternate solution just because i found other solutions as 'complicated or long or i wanted a shortcut method or solutions looks hard', i actually solved this question and arrived at the correct answer in the same way as you and others did. The reason i asked for a alternate solution is because as we all know Bunuel and other experts too often come up with a solution which is completely different and less time consuming. That was exactly my reason behind asking for an alternate solution. Also the post was around 3 years old so was hoping if someone may have some new information on this. Probability is one of the challenging topics for me, even though i was able to solve this problem i felt i took more time to solve it than that is necessary and i just wanted to improve on it by taking some help. Thanks again for your advise and insights on facing problems like these.



Intern
Joined: 25 Dec 2016
Posts: 21
Location: India

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
05 Jan 2017, 10:38
Here is the method that I used.
1) Number of outcomes with 6T = 1 2) Number of outcomes with 5T +1H = 6!/5! = 6 3) Number of outcomes with 4T +2H = 6!/(4!*2!) = 15 [But this includes combinations where the 2Hs are together] i.e. [HH] is placed in any of the spaces (_) in _T_T_T_T_ So, subtracting 5 from (3). 4) Number of outcomes with 3T +3H = 6!/(3!*3!) = 20 [Here again, we have to subtract combinations where the 2Hs are together] i.e. [HH] is placed in any of the spaces (_) in {[_T_T_T_H], [_T_T_HT_], [_T_HT_T_], [_HT_T_T_]}. Realize I've removed one space (_) after the each H. So, subtracting 4*4 = 16. Above will take care of all combinations where 3Hs are together.
Total favourable combinations = 1+6+155+2016 = 21 P = 21/64 A.



NonHuman User
Joined: 09 Sep 2013
Posts: 9184

Re: A fair coin is tossed 6 times. What is the probability of getting no
[#permalink]
Show Tags
01 Jul 2018, 03:33
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: A fair coin is tossed 6 times. What is the probability of getting no &nbs
[#permalink]
01 Jul 2018, 03:33






