Let's analyze the situation mathematically to determine who is correct about the probabilities of getting all questions correct versus getting exactly one question correct on a four-question true-false quiz.
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Probability of getting all correct answers: For each question, the probability of guessing correctly is \( \frac{1}{2} \). If there are 4 questions, the probability \( P(\text{all correct}) \) is: \[ P(\text{all correct}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]
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Probability of getting exactly one correct answer: The probability of getting exactly one question correct involves calculating the combination of selecting 1 question from 4 to be correct (and the others incorrect). This can be calculated as follows:
- The probability of getting one specific question correct is \( \frac{1}{2} \) and the probability of getting three questions incorrect is \( \left(\frac{1}{2}\right)^3 \).
- The number of ways to choose which one of the four questions is correct is given by \( \binom{4}{1} \).
Therefore, the probability \( P(\text{exactly one correct}) \) is: \[ P(\text{exactly one correct}) = \binom{4}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^3 = 4 \cdot \frac{1}{2} \cdot \frac{1}{8} = 4 \cdot \frac{1}{16} = \frac{4}{16} = \frac{1}{4} \]
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Comparison of probabilities:
- The probability of getting all correct: \( \frac{1}{16} \)
- The probability of getting exactly one correct: \( \frac{1}{4} \) or \( \frac{4}{16} \)
Since \( \frac{1}{4} \) (the probability of getting exactly one correct answer) is greater than \( \frac{1}{16} \) (the probability of getting all correct answers), we find that Alex is correct in his belief that the probability of getting exactly one correct answer is greater than the probability of getting all correct answers.
Therefore, the answer is Alex is correct.
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