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PP1: A contest consists of n questions, each answered

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Guest
 Post subject: PP1: A contest consists of n questions, each answered
 Post Posted: Tue May 13, 2008 4:48 pm 
A contest consists of n questions, each answered either True or False. Anyone who answers all n correctly will be a winner. What is the least value of n for which the probability is Less than 1/ 1000 that a person who randomly guesses the answer to each will be a winner.

Answer: 10.

Had absolutely no clue where to begin with this one...no probability theory i could think to apply helped in any way!
Simply stuck!

Thanks
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RonPurewal
 Post subject: Re: PP1: A contest consists of n questions, each answered
 Post Posted: Wed May 14, 2008 5:48 am 
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ManhattanGMAT Staff


Posts: 8087
Anonymous wrote:
A contest consists of n questions, each answered either True or False. Anyone who answers all n correctly will be a winner. What is the least value of n for which the probability is Less than 1/ 1000 that a person who randomly guesses the answer to each will be a winner.

Answer: 10.

Had absolutely no clue where to begin with this one...no probability theory i could think to apply helped in any way!
Simply stuck!

Thanks


rephrase:
what is the least value of n for which there is less than a 1/1000 chance of guessing n questions in a row correctly?'

here's the deal:
* there is a 1/2 chance of guessing each question correctly
* each question is independent of the other questions, so the chance of guessing n questions correctly is (1/2)(1/2)(1/2)...(1/2), where there are n (1/2)'s
* this is (1/2)^n, or 1/(2^n)

so:
1/2^n < 1/1000
take reciprocals:
2^n > 1000
n > 10 (because 2^10 = 1024)

i've seen big powers of two in gmatprep problems before, but 2^10 is definitely the biggest i've yet seen.
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jyothi.11182
 Post subject: Re: PP1: A contest consists of n questions, each answered
 Post Posted: Fri Aug 17, 2012 11:26 pm 
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Students


Posts: 1
Hi

I had a small doubt on the above expln. No where in the question does he mention that true and false are equally likely. So we jus assume that they are equally probable and go ahead and take the probablity to be 1/2 ?

Thanks
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jnelson0612
 Post subject: Re: PP1: A contest consists of n questions, each answered
 Post Posted: Sun Aug 19, 2012 3:41 pm 
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ManhattanGMAT Staff


Posts: 2390
jyothi.11182 wrote:
Hi

I had a small doubt on the above expln. No where in the question does he mention that true and false are equally likely. So we jus assume that they are equally probable and go ahead and take the probablity to be 1/2 ?

Thanks


The question says that the contestant will "randomly" guess answers to the questions. When I randomly guess answers I have an equal chance of getting each answer. For example, if I took the GMAT but just picked the first letter I felt like picking for each question, I would have a 1/5 chance of answering correctly. Similarly a person "randomly" answering (or in other words, not trying at all, just picking a "random" answer) in this scenario would have a 1/2 chance of getting each question correct.

_________________
Jamie Nelson
ManhattanGMAT Instructor
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