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# A contest will consist of n questions, each of which is to

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VP
Joined: 21 Jul 2006
Posts: 1363
A contest will consist of n questions, each of which is to  [#permalink]

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21 Nov 2007, 16:13
1
16
00:00

Difficulty:

35% (medium)

Question Stats:

73% (01:00) correct 27% (01:28) wrong based on 550 sessions

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A contest will consist of n questions, each of which is to be answered either "True" or "False." Anyone who answers all n questions 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 question will be winner?

a) 5
b) 10
c) 50
d) 100
e) 1000

OPEN DISCUSSION OF THIS QUESTION IS HERE: a-contest-will-consist-of-n-questions-each-of-which-is-to-38710.html
Director
Joined: 03 Sep 2006
Posts: 769

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21 Nov 2007, 19:20
8
Each question has the probability of being correct equal to " 1/2 ".

So the question is basically asking us:

(1/2)^n <1> 1000

n = 10 gives us 1024, which is greater than 1000.

Thus answer should be n = 10. The choice "b".
Manager
Joined: 03 Sep 2006
Posts: 231

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21 Nov 2007, 19:39
1
15 sec. question.
The same approach as abhishakes2001
Senior Manager
Joined: 02 Dec 2007
Posts: 382

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23 Sep 2008, 11:36
tarek99 wrote:
A contest will consist of n questions, each of which is to be answered either "True" or "False." Anyone who answers all n questions 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 question will be winner?

a) 5
b) 10
c) 50
d) 100
e) 1000

B

2^(-n) > 10^-3 => 2^-3 *5^-3 => 2^-3*(4)^-3 => 2^-9

so n > -9
Manager
Joined: 18 Jun 2007
Posts: 247

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23 Sep 2008, 12:59
interesting tarek, this is GMAT Prep 2 question.
I got this question as 37th question me too got it wrong but then Probability has never been my strong point but still it is interesting cause got this '15 sec'. question on 37th number though I scored 760 in Prep2.
Director
Joined: 25 Oct 2008
Posts: 500
Location: Kolkata,India

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24 Jul 2009, 18:53
1
i was looking for a way out without calculating 2^10
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Manager
Joined: 22 Jul 2009
Posts: 161

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29 Aug 2009, 16:03
2
tejal777 wrote:
i was looking for a way out without calculating 2^10

Manually. Get 2. 2^2=4. Multiply that 8 times

Shortcut: remember that 2^5=32. Then multiply 5 times.
32x2=64=2^6
64x2=128=2^7
128x2=256=2^8
256x2=512=2^9
512x2=1024=2^10
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Manager
Joined: 14 Aug 2009
Posts: 115

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30 Aug 2009, 07:13
2
2^10 in computer science is K.
for instance, 1MB=1024KB=2^10KB

a very useful number which should be kept in mind.
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Manager
Joined: 27 Oct 2008
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27 Sep 2009, 02:12
(1/2)^10 < 1/1000

Hence 10 questions is the minimum
Manager
Joined: 02 Jan 2010
Posts: 103

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08 Sep 2010, 03:20
wow nice one!!!!abhishakes2001 nailed it
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Ganesh
Class of 2012
Great Lakes Institute of Management
http://greatlakes.edu.in
Manager
Joined: 08 Sep 2010
Posts: 121

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17 Jun 2011, 19:54
agree with 10...maybe a 40 sec question
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Intern
Joined: 23 May 2011
Posts: 7

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17 Jun 2011, 21:29
1
P(T)=1/2
P(F)=1/2

P(n)= Favourable/ Total
= 1/ 2^n

For 1/2^n <1/1000

least value of n is 10.

Hope this helps.
Math Expert
Joined: 02 Sep 2009
Posts: 55266
Re: A contest will consist of n questions, each of which is to  [#permalink]

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11 Sep 2013, 11:16
7
1
A contest will consist of n questions, each of which is to be answered either "True" or "False." Anyone who answers all n questions 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 question will be winner?

A. 5
B. 10
C. 50
D. 100
E. 1000

The probability to randomly guess the answer for 1 question is 1/2, for 2 questions 1/2*1/2=1/2^2, similarly for n questions the probability is 1/2^n.

We need to find the least value of $$n$$ for which $$\frac{1}{2^n}<\frac{1}{1,000}$$ --> $$2^n>1,000$$ --> $$n_{min}=10$$.

OPEN DISCUSSION OF THIS QUESTION IS HERE: a-contest-will-consist-of-n-questions-each-of-which-is-to-38710.html
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Re: A contest will consist of n questions, each of which is to  [#permalink]

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15 Sep 2013, 06:39
tarek99 wrote:
A contest will consist of n questions, each of which is to be answered either "True" or "False." Anyone who answers all n questions 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 question will be winner?
a) 5
b) 10
c) 50
d) 100
e) 1000

In all there are n questions, and each question can be answered 2 ways (i.e. True or False). So we will get $$2^n$$ different sequences of answers. Of which one sequence is TTTTTTTT......n times (i.e. All correct Answers)

We are told that The person who get all the answers correct that means who get the sequence mentioned above (TTTTT.... n times) will be a winner.

A person can choose any sequence from $$2^n$$ sequences. He has to choose TTTT.... n times in order to win the game.

So Probability that a person will win the game is $$\frac{1}{2^n}$$

What is the least value of n for which the probability is less than 1/1000 -------> $$\frac{1}{2^n} < \frac{1}{1000}$$ -----------------------> Here we can cross multiply the inequality since we know that $$2^n$$ will always be positive (We know that n can neither be zero nor be Negative)

So We have that $$2^n > 1000$$ --------> Using n=10 we get $$1024 > 1000$$ Sufficient.
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Re: A contest will consist of n questions, each of which is to  [#permalink]

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25 Aug 2018, 06:03
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Re: A contest will consist of n questions, each of which is to   [#permalink] 25 Aug 2018, 06:03
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