Question

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?

Answer 1

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

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