Probability

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mgmt_gmat: Master | Next Rank: 500 Posts; Posts: 221; Joined: Mon Dec 22, 2008 6:24 pm; Thanked: 2 times

by mgmt_gmat » Thu Feb 11, 2010 5:41 am

n questions can either be true or false. If you answer all n correct you win. What is the least value of n for which the probability is less than 1/1000 for you to win by guessing randomly?
a. 5
b. 10
c. 50
d. 100
e. 1000

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Source: Beat The GMAT — Problem Solving | Thu Feb 11, 2010 5:41 am

dtweah: Master | Next Rank: 500 Posts; Posts: 487; Joined: Fri Mar 27, 2009 5:49 am; Thanked: 36 times

by dtweah » Thu Feb 11, 2010 6:27 am

mgmt_gmat wrote:n questions can either be true or false. If you answer all n correct you win. What is the least value of n for which the probability is less than 1/1000 for you to win by guessing randomly?
a. 5
b. 10
c. 50
d. 100
e. 1000

An easy way to see this is to consider the problem as a toss of a fair coin. Each side has probability 1/2 of occuring. If we toss a coin one time, we have as HEAD=Correct answers the probability 1/2. If we toss it twice we have probability (1/2)^2=1/4. So n is the number of questions on the exam. If there are only 2 questions, we win with the same probability of 1/4. Suppose there are n questions. Then we have (1/2)^n. or 1/2^n. This problem is testing your knowledge of powers of 2 up to 10. It is useful to know 2^10=1024. So you are just comparing 1/1024 and 1/1000. So when n =10 we get a probability that is less than 1/1000 for the first time. All previous n's will give greater probabilities. For example 1/2^5 is 1/32> 1/1000. Of course, any n coming after 10 makes the fraction even smaller But the question wants the SMALLEST of Such N's.
Choose B.

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thephoenix: Legendary Member; Posts: 1560; Joined: Tue Nov 17, 2009 2:38 amFollowed by: 5 members; Thanked: 137 times

by thephoenix » Thu Feb 11, 2010 6:27 am

IMO 10

for a q to be correct P=1/2 (T or F)=P of win
for n Q P of win=(1/2)^n

now for (1/2)^n<1/1000--->2^n>1000---> for it to be true n>=10

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shashank.ism: Legendary Member; Posts: 1022; Joined: Mon Jul 20, 2009 11:49 pmFollowed by: 2 members; Location: Gandhinagar; Thanked: 41 times

by shashank.ism » Thu Feb 11, 2010 7:01 am

mgmt_gmat wrote:n questions can either be true or false. If you answer all n correct you win. What is the least value of n for which the probability is less than 1/1000 for you to win by guessing randomly?
a. 5
b. 10
c. 50
d. 100
e. 1000

Since a question can be either true or false. So probability of getting true or false = 1/2
i.e., P(true)= 1/2, P(false) =1/2
now p( n all correct) = (1/2)^n <1/1000 --> 2^n > 1000
min value = 2^10 ==> n =10 Ans B

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ajith: Legendary Member; Posts: 1275; Joined: Thu Sep 21, 2006 11:13 pmFollowed by: 2 members; Location: Arabian Sea; Thanked: 125 times

by ajith » Thu Feb 11, 2010 8:13 am

mgmt_gmat wrote:n questions can either be true or false. If you answer all n correct you win. What is the least value of n for which the probability is less than 1/1000 for you to win by guessing randomly?
a. 5
b. 10
c. 50
d. 100
e. 1000

Probability of getting 1 question right = 1/2
Probability of getting 2 questions right on a row = 1/2*1/2 = 1/2^2
Probability of getting 3 questions right on a row = 1/2*1/2*1/2 = 1/2^3
Probability of getting n questions right on a row = 1/2^n

1/2^n<1/1000

1/2^9 = 1/512
1/2^10 =1/1024

Hence n = 10

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ganesh prasath: Junior | Next Rank: 30 Posts; Posts: 14; Joined: Sun Sep 12, 2010 9:09 pm

by ganesh prasath » Sun Jun 05, 2011 5:05 am

y not 2^1000? that ll definitely give the least value right??

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cans: Legendary Member; Posts: 1309; Joined: Mon Apr 04, 2011 5:34 amFollowed by: 123 members; Location: India; Thanked: 310 times; GMAT Score:750

by cans » Sun Jun 05, 2011 5:20 am

ganesh prasath wrote:y not 2^1000? that ll definitely give the least value right??

Probability of getting all correct = (1/2)^n = 1/(2^n)
as this has to be <1/1000
it means 1/(2^n)<1/1000
2^n>1000
2^9=512
2^10=1024
thus min is 10

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sandeep800: Master | Next Rank: 500 Posts; Posts: 126; Joined: Tue Feb 09, 2010 6:37 pmFollowed by: 4 members

by sandeep800 » Sun Jun 05, 2011 10:36 am

Thanx Cans

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