To determine the probability that a student answers all the questions correctly by guessing, we can follow these steps:
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Identify the number of questions and choices per question:
- The quiz has \(4\) questions.
- Each question has \(5\) choices.
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Calculate the probability of guessing one question correctly: The probability \(P\) of guessing one question correctly is: \[ P(\text{correct}) = \frac{1}{\text{number of choices}} = \frac{1}{5} \]
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Calculate the probability of answering all questions correctly: Since the questions are independent (the outcome of one does not affect the others), we can multiply the probabilities of guessing each question correctly: \[ P(\text{all correct}) = P(\text{correct})^n = \left(\frac{1}{5}\right)^n \] where \(n\) is the number of questions. Here, \(n = 4\).
Therefore, \[ P(\text{all correct}) = \left(\frac{1}{5}\right)^4 = \frac{1}{625} \]
So, the probability that the student answers all the questions correctly is
\[ \frac{1}{625}. \]