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PP1: A contest consists of n questions, each answered

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 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: 7146
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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