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
Answer Choice BIn 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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