A math quiz has

4
4 multiple choice questions and each question has
5
5 choices. If a student has to guess (and is not able to remove any of the incorrect choices) what is the probability that the student answers all the questions correctly?

1 answer

To determine the probability that a student answers all the questions correctly by guessing, we can follow these steps:

  1. Identify the number of questions and choices per question:

    • The quiz has \(4\) questions.
    • Each question has \(5\) choices.
  2. Calculate the probability of guessing one question correctly: The probability \(P\) of guessing one question correctly is: \[ P(\text{correct}) = \frac{1}{\text{number of choices}} = \frac{1}{5} \]

  3. 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

\[ \frac{1}{625}. \]