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# If an unbiased coin is flipped 5 times, the probability that

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Joined: 08 Sep 2010
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If an unbiased coin is flipped 5 times, the probability that  [#permalink]

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Updated on: 28 Oct 2010, 03:43
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Difficulty:

95% (hard)

Question Stats:

31% (02:27) correct 69% (02:29) wrong based on 152 sessions

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If an unbiased coin is flipped 5 times, the probability that the same face does not show up in any three consecutive flips is

A. 1/2
B. 5/8
C. 3/8
D. 7/8
E. 9/8

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Originally posted by ankitranjan on 28 Oct 2010, 02:41.
Last edited by ankitranjan on 28 Oct 2010, 03:43, edited 1 time in total.
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Re: If an unbiased coin is flipped..... Help Needed  [#permalink]

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28 Oct 2010, 03:33
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1
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ankitranjan wrote:
If an unbiased coin is flipped 5 times,the probabilty that the same face does not show up in any three consecutive flips is

A. 1/2
B. 5/8
C. 3/8
D. 7/8
E. 9/8

Probably the easiest way would be just to write down the cases with at least 3 consecutive faces:

HHHHH
HHHHT
HHHTH
HHHTT

TTHHH
THHHH
HTHHH

THHHT

So we have 8 cases for at least 3 consecutive heads, which means that there will be also 8 cases for at least 3 consecutive tails. As there are total of 2^5 outcomes then the probability would be $$P=\frac{8+8}{2^5}=\frac{1}{2}$$.

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Manager
Joined: 08 Sep 2010
Posts: 148
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Re: If an unbiased coin is flipped..... Help Needed  [#permalink]

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28 Oct 2010, 03:45
Bunuel that was the right answer.Thanks for the explanation.
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Re: If an unbiased coin is flipped..... Help Needed  [#permalink]

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10 Nov 2010, 04:52
Bunuel, i know it will be the same answer, but shouldnt it be 1-p here?
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Re: If an unbiased coin is flipped..... Help Needed  [#permalink]

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10 Nov 2010, 04:57
144144 wrote:
Bunuel, i know it will be the same answer, but shouldnt it be 1-p here?

Yes (as we counted the probability of opposite event).
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Re: If an unbiased coin is flipped..... Help Needed  [#permalink]

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10 Nov 2010, 07:48
ankitranjan wrote:
If an unbiased coin is flipped 5 times,the probabilty that the same face does not show up in any three consecutive flips is

A. 1/2
B. 5/8
C. 3/8
D. 7/8
E. 9/8

I agree with Bunuel. The easiest solution I could think of was enumerating in a sequence. I try and find how I can have the same face three or more times.

Start with 3 Hs and 2 Ts, go on to 4Hs and 1T and finally 5Hs
The 3 Hs have to be together so: HHHTT, TTHHH, THHHT
Then, 4 Hs and 1T. Place the 4 Hs first: H H H H. Now look for the places where you can place the T: THHHH, HTHHH, HHHTH and HHHHT
Finally 5 Hs: HHHHH
Total 8 ways to get 3 or more Heads in a row. Ways to get 3 or more Tails in a row is also 8.
Total number of ways outcomes with 5 coins = 2^5 = 32
Probability of same face occurring 3 or more times = 16/32 = 1/2
Probability of same face occurring less than 3 times = 1 - 1/2 = 1/2
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Re: If an unbiased coin is flipped..... Help Needed  [#permalink]

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09 Nov 2013, 00:19
VeritasPrepKarishma wrote:
ankitranjan wrote:
If an unbiased coin is flipped 5 times,the probabilty that the same face does not show up in any three consecutive flips is

A. 1/2
B. 5/8
C. 3/8
D. 7/8
E. 9/8

I agree with Bunuel. The easiest solution I could think of was enumerating in a sequence. I try and find how I can have the same face three or more times.

Start with 3 Hs and 2 Ts, go on to 4Hs and 1T and finally 5Hs
The 3 Hs have to be together so: HHHTT, TTHHH, THHHT
Then, 4 Hs and 1T. Place the 4 Hs first: H H H H. Now look for the places where you can place the T: THHHH, HTHHH, HHHTH and HHHHT
Finally 5 Hs: HHHHH
Total 8 ways to get 3 or more Heads in a row. Ways to get 3 or more Tails in a row is also 8.
Total number of ways outcomes with 5 coins = 2^5 = 32
Probability of same face occurring 3 or more times = 16/32 = 1/2
Probability of same face occurring less than 3 times = 1 - 1/2 = 1/2

Is there a straighter way to solve this problem? Without going the opposite way
My attempt: Of the five slots, three straight slots are reserved for opposite results. (HTH, THT). The remaining two slots can yield a head or a tail.
So the possible number of outcomes is 2*2 = 4. Now these two slots are always clubbed together and can be treated as one slot that can be adjusted anywhere among the 4 possible slots in 4c1 ways. So the total possible outcomes under the given conditions will be 2*2* 4c1 = 16

Probability is 16/32 = 1/2

Is the thinking behind reaching this solution correct? Or did I just get lucky with the answer?

TIA
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Re: If an unbiased coin is flipped 5 times, the probability that  [#permalink]

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12 Oct 2018, 22:10
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Re: If an unbiased coin is flipped 5 times, the probability that   [#permalink] 12 Oct 2018, 22:10
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