Que: A quiz consists of X questions, each of which is to be answered either “Yes” or “No.” What is the least value of X for which the probability is less than \(\frac{1}{500}\) such that a participant who randomly guesses the answer to each question will be a winner?
(A) 8
(B) 9
(C) 10
(D) 200
(E) 500
Que: A quiz consists of X questions, each of which is to be answered either “Yes” or “No.” What is the least value....
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- Max@Math Revolution
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Solution: Total questions: X
Total options for each question: 2 [Yes or No]
Thus, the probability of randomly guessing an answer and getting it correct = \(\frac{1}{2}\)
Thus, the probability of randomly guessing answers to all X questions and getting them correct:
=> \(\frac{1}{2}\) * \(\frac{1}{2}\) * ….. X times
=> \(\left(\frac{1}{2}\right)^x\)
=> \(\left(\frac{1}{2}\right)^x\ <\frac{1}{500}\)
=> \(\left(\frac{1}{2^x}\right)\ <\frac{1}{500}\)
=> \(2^x>500\)
For x = 9, \(2^9=512\), which just exceeds 500.
Therefore, B is the correct answer.
Answer B
Total options for each question: 2 [Yes or No]
Thus, the probability of randomly guessing an answer and getting it correct = \(\frac{1}{2}\)
Thus, the probability of randomly guessing answers to all X questions and getting them correct:
=> \(\frac{1}{2}\) * \(\frac{1}{2}\) * ….. X times
=> \(\left(\frac{1}{2}\right)^x\)
=> \(\left(\frac{1}{2}\right)^x\ <\frac{1}{500}\)
=> \(\left(\frac{1}{2^x}\right)\ <\frac{1}{500}\)
=> \(2^x>500\)
For x = 9, \(2^9=512\), which just exceeds 500.
Therefore, B is the correct answer.
Answer B
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