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A fair coin is tossed 6 times

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A fair coin is tossed 6 times [#permalink] New post 23 Sep 2012, 02:34
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A
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C
D
E

Difficulty:

  55% (hard)

Question Stats:

27% (01:25) correct 73% (02:02) wrong based on 33 sessions
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
[Reveal] Spoiler: OA

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Re: A fair coin is tossed 6 times [#permalink] New post 24 Sep 2012, 21:34
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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 pain-in-the-tush than what you will see on the GMAT. I don't know the source, but this seems to come from some over-achieving 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/gmat-quant-how-to-count/

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 tails

For this case, obviously you can't have two heads in a row. There's only one way this can happen: TTTTTT
ONE

Case Two: five tails, one head
Again, 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)
SIX

Case Three: four tails, two heads
This 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.
TEN

Case Four: three tails, three heads
Now, 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

Add those up: 1 + 6 + 10 + 2 = 19. That's our numerator.

Probability = 19/64
Answer = C

Does all that make sense? Please let me know if you have any further questions.

Mike :-)
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Re: A fair coin is tossed 6 times [#permalink] New post 24 Sep 2012, 21:57
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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}

Last edited by MonSama on 24 Sep 2012, 22:02, edited 3 times in total.
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Re: A fair coin is tossed 6 times [#permalink] New post 24 Sep 2012, 21:59
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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 heads
Now, 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
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Re: A fair coin is tossed 6 times [#permalink] New post 25 Sep 2012, 08:19
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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 :-)
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Re: A fair coin is tossed 6 times [#permalink] New post 25 Sep 2012, 23:09
Thanks for the comprehensive solution to both Mike and Monsama .. Kudos for both of you !!
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Re: A fair coin is tossed 6 times [#permalink] New post 26 Sep 2012, 00:24
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All possibilities with no 2 H touching each other.
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Re: A fair coin is tossed 6 times [#permalink] New post 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
a-fair-coin-is-tossed-5-times-what-is-the-probability-of-99478.html
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Re: A fair coin is tossed 6 times   [#permalink] 31 Jul 2014, 14:59
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