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# If a child flips a coin five times in a row, what is the probability

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Intern
Joined: 06 Apr 2015
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If a child flips a coin five times in a row, what is the probability  [#permalink]

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17 Dec 2015, 13:24
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If a child flips a coin five times in a row, what is the probability that she will receive at least one head and one tail?

A) 3/4
B) 11/12
C) 15/16
D) 31/32
E) 63/64

Guys please help me. i know only one way which is a really long method wherein you list all possibilities but given the time constraints, its not possible. please show a shorter and easy way to solve this please
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Re: If a child flips a coin five times in a row, what is the probability  [#permalink]

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17 Dec 2015, 21:03
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nikdiablo129 wrote:
If a child flips a coin five times in a row, what is the probability that she will receive at least one head and one tail?

A) 3/4
B) 11/12
C) 15/16
D) 31/32
E) 63/64

Guys please help me. i know only one way which is a really long method wherein you list all possibilities but given the time constraints, its not possible. please show a shorter and easy way to solve this please

The coin is flipped 5 times in a row.
So there are 2*2*2*2*2 = 32 different outcomes possible such as HHTHH, HTHHT, TTHTH etc

Look for the reverse probability - when will there is no heads or no tails?
There will be no heads when you get all tails. There is only 1 such case TTTTT
There will be no tails when you get all heads. There is again only 1 such case HHHHH
So, out of 32, there are only 2 cases in which there are either no heads or no tails. In all other 30 cases, there is at least one heads and at least one tails.

Required probability = 30/32 = 15/16

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Karishma
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Save up to $1,000 on GMAT prep through 8/20! Learn more here > GMAT self-study has never been more personalized or more fun. Try ORION Free! ##### General Discussion Math Expert Joined: 02 Aug 2009 Posts: 6537 Re: If a child flips a coin five times in a row, what is the probability [#permalink] ### Show Tags 17 Dec 2015, 21:18 nikdiablo129 wrote: If a child flips a coin five times in a row, what is the probability that she will receive at least one head and one tail? A) 3/4 B) 11/12 C) 15/16 D) 31/32 E) 63/64 Guys please help me. i know only one way which is a really long method wherein you list all possibilities but given the time constraints, its not possible. please show a shorter and easy way to solve this please Hi, you don't require to write down all possiblities but do the reverse as also suggested by karishma.. there are two ways where we do not have atleast one each of heads and tail and these are HHHHH and TTTTT.. prob of HHHHH=$$1/2*1/2*1/2*1/2*1/2=\frac{1}{32}$$... similarly for TTTTT= 1/32.. therefore the prob of not getting one of each =1/32+1/32=1/16.. therefore prob of getting atleast one of each=1-1/16=15/16.. _________________ 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 Intern Joined: 06 Apr 2015 Posts: 4 Re: If a child flips a coin five times in a row, what is the probability [#permalink] ### Show Tags 18 Dec 2015, 02:43 VeritasPrepKarishma wrote: nikdiablo129 wrote: If a child flips a coin five times in a row, what is the probability that she will receive at least one head and one tail? A) 3/4 B) 11/12 C) 15/16 D) 31/32 E) 63/64 Guys please help me. i know only one way which is a really long method wherein you list all possibilities but given the time constraints, its not possible. please show a shorter and easy way to solve this please The coin is flipped 5 times in a row. So there are 2*2*2*2*2 = 32 different outcomes possible such as HHTHH, HTHHT, TTHTH etc Look for the reverse probability - when will there is no heads or no tails? There will be no heads when you get all tails. There is only 1 such case TTTTT There will be no tails when you get all heads. There is again only 1 such case HHHHH So, out of 32, there are only 2 cases in which there are either no heads or no tails. In all other 30 cases, there is at least one heads and at least one tails. Required probability = 30/32 = 15/16 Answer (C) Thank you so much maam. I give my gmat 2nd attempt on 22nd i have been practicing the 6 tests from Veritas prep Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8188 Location: Pune, India Re: If a child flips a coin five times in a row, what is the probability [#permalink] ### Show Tags 18 Dec 2015, 21:25 nikdiablo129 wrote: Thank you so much maam. I give my gmat 2nd attempt on 22nd i have been practicing the 6 tests from Veritas prep Wishing you the best! _________________ Karishma Veritas Prep GMAT Instructor Save up to$1,000 on GMAT prep through 8/20! Learn more here >

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Re: If a child flips a coin five times in a row, what is the probability  [#permalink]

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20 Dec 2015, 20:02
Hi nikdiablo129,

When dealing with probabilities, there are two results that can be calculated: what you WANT to have happen and what you DON'T WANT to have happen. In certain questions, it's actually faster to calculate what you DON'T WANT, since....

(Probability of what you WANT) + (Probability of what you DON'T WANT) = 1

Keep an eye out for the second option. Calculating what you DON'T WANT and then subtracting that fraction from 1 might be the faster/easier option.

GMAT assassins aren't born, they're made,
Rich
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Co-Founder & GMAT Assassin

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!*********************** Manager Joined: 15 May 2010 Posts: 173 Location: India Concentration: Strategy, General Management WE: Engineering (Manufacturing) Re: If a child flips a coin five times in a row, what is the probability [#permalink] ### Show Tags 26 Dec 2015, 01:06 VeritasPrepKarishma wrote: nikdiablo129 wrote: If a child flips a coin five times in a row, what is the probability that she will receive at least one head and one tail? A) 3/4 B) 11/12 C) 15/16 D) 31/32 E) 63/64 Guys please help me. i know only one way which is a really long method wherein you list all possibilities but given the time constraints, its not possible. please show a shorter and easy way to solve this please The coin is flipped 5 times in a row. So there are 2*2*2*2*2 = 32 different outcomes possible such as HHTHH, HTHHT, TTHTH etc Look for the reverse probability - when will there is no heads or no tails? There will be no heads when you get all tails. There is only 1 such case TTTTT There will be no tails when you get all heads. There is again only 1 such case HHHHH So, out of 32, there are only 2 cases in which there are either no heads or no tails. In all other 30 cases, there is at least one heads and at least one tails. Required probability = 30/32 = 15/16 Answer (C) Teacher, I have one query. The question writes to find the probability of at least 0ne head and one tail. I understood the process explained above. But the confusion arises from the word 'And' between two events. As per concept, “AND” means MULTIPLICATION of two independent events. Please clear my doubts. Regards, Sun01 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8188 Location: Pune, India Re: If a child flips a coin five times in a row, what is the probability [#permalink] ### Show Tags 27 Dec 2015, 20:21 sun01 wrote: VeritasPrepKarishma wrote: nikdiablo129 wrote: If a child flips a coin five times in a row, what is the probability that she will receive at least one head and one tail? A) 3/4 B) 11/12 C) 15/16 D) 31/32 E) 63/64 Guys please help me. i know only one way which is a really long method wherein you list all possibilities but given the time constraints, its not possible. please show a shorter and easy way to solve this please The coin is flipped 5 times in a row. So there are 2*2*2*2*2 = 32 different outcomes possible such as HHTHH, HTHHT, TTHTH etc Look for the reverse probability - when will there is no heads or no tails? There will be no heads when you get all tails. There is only 1 such case TTTTT There will be no tails when you get all heads. There is again only 1 such case HHHHH So, out of 32, there are only 2 cases in which there are either no heads or no tails. In all other 30 cases, there is at least one heads and at least one tails. Required probability = 30/32 = 15/16 Answer (C) Teacher, I have one query. The question writes to find the probability of at least 0ne head and one tail. I understood the process explained above. But the confusion arises from the word 'And' between two events. As per concept, “AND” means MULTIPLICATION of two independent events. Please clear my doubts. Regards, Sun01 AND is one of the most commonly used words and it is not possible that it should be used only when you need to multiply the probability of two independent events. The concept is that if there are two independent events and you need to find the probability that one happens "and" the other happens, then you multiply the probabilities. Note that the actual question may not even use "and". It may use "as well as" or "additionally" etc. Focus on the logic of the question. _________________ Karishma Veritas Prep GMAT Instructor Save up to$1,000 on GMAT prep through 8/20! Learn more here >

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Manager
Joined: 25 Dec 2012
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Re: If a child flips a coin five times in a row, what is the probability  [#permalink]

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06 Jan 2016, 08:45
In probability, whenever you see "ATLEAST" or "ATMOST" in the question stem choose for the reverse option. That will make things simpler.

If question is saying at least one head in 3 flips means the reverse is zero heads which is all tails.

Hope this clears.
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Joined: 08 Dec 2015
Posts: 302
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If a child flips a coin five times in a row, what is the probability  [#permalink]

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18 Jun 2016, 04:33
VeritasPrepKarishma

What if we convert the H-T event into a single event?

Then we would get 4 possible rolls and we could subtract 1-P(of HT not happening) that would be 16/16-1/16?

Like this makes sense, but for me there is a problem to understand what exactly "HT not happening" means in the new context of 4 rolls and HT being a single event.

Can this technique be used here? Or its too confusing?

Thank you!
Veritas Prep GMAT Instructor
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Re: If a child flips a coin five times in a row, what is the probability  [#permalink]

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19 Jun 2016, 22:30
iliavko wrote:
VeritasPrepKarishma

What if we convert the H-T event into a single event?

Then we would get 4 possible rolls and we could subtract 1-P(of HT not happening) that would be 16/16-1/16?

Like this makes sense, but for me there is a problem to understand what exactly "HT not happening" means in the new context of 4 rolls and HT being a single event.

Can this technique be used here? Or its too confusing?

Thank you!

Yes, the logic is that if you have at least one heads and at least one tails, there will be at least one pair of HT or TH when you write down the 5 rolls.

e.g. HHHHT
THHTH
etc

Every pair is either HH, TT, HT or TH so probability of HT or TH is 1/2.

Starting from the first roll, in HHHHT, you get 4 pairs: HH, HH, HH, HT

Probability that all for pairs are HH or TT is 1/2 * 1/2 * 1/2 * 1/2 = 1/16

Probability that you have at least one HT or TH pair = 1 - 1/16 = 15/16

But this method just complicates a simple question.
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Save up to $1,000 on GMAT prep through 8/20! Learn more here > GMAT self-study has never been more personalized or more fun. Try ORION Free! Senior Manager Joined: 08 Dec 2015 Posts: 302 GMAT 1: 600 Q44 V27 Re: If a child flips a coin five times in a row, what is the probability [#permalink] ### Show Tags 20 Jun 2016, 10:52 Thank you so much, Karishima! As you said, indeed the conversion to a single event only complicated this problem! Thank you for your input Intern Joined: 19 Jan 2016 Posts: 47 Re: If a child flips a coin five times in a row, what is the probability [#permalink] ### Show Tags 03 Jun 2017, 05:01 i tried solving the problem by using the formula : - Total = All heads + (At least 1 tail), the following equation resulted in 31/32 Probability of all heads = 1, so at least 1 tail would be 1- 1/32 = 31/32. (at least 1 tail) Similarly for heads Though i did guess the right answer i would like to know whats wrong with my approach EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12185 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If a child flips a coin five times in a row, what is the probability [#permalink] ### Show Tags 03 Jun 2017, 12:33 Hi anuj11, I think that there might be some 'typos' in what you wrote. When flipping a coin 5 times, there are only 2 outcomes that do not include at least one head and at least one tail.... To start, the probability of getting "all heads" is (1/2)(1/2)(1/2)(1/2)(1/2) = 1/32 and the same probability exists for "all tails" (re: 1/32). Thus, the probability of "all heads" or "all tails" = (1/32) + (1/32) = 2/32 = 1/16 Taking all of this one step further, the probability of getting "AT LEAST one head AND AT LEAST one tail" = 1 - 1/16 = 15/16 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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