To determine the probability that Josiah will get all three questions correct on a multiple-choice quiz where each question has five possible answer options, we need to consider the probability of answering each individual question correctly and then use the fact that the events are independent (since each question is a separate event).
### Step-by-Step Solution:
1. Calculate the Probability of Answering One Question Correctly:
Since there are five answer options for each question and only one of them is correct, the probability of Josiah choosing the correct answer for a single question is:
[tex]\[
\text{Probability of a single correct answer} = \frac{1}{5}
\][/tex]
2. Determine the Probability of Answering All Three Questions Correctly:
Because Josiah’s choice for each question is independent of his choices for the other questions, we multiply the probabilities of each event happening. Thus, the probability of getting all three questions correct is:
[tex]\[
\text{Probability of all three correct answers} = \left(\frac{1}{5}\right)^3
\][/tex]
3. Perform the Multiplication:
Evaluate [tex]\(\left(\frac{1}{5}\right)^3\)[/tex]:
[tex]\[
\left(\frac{1}{5}\right)^3 = \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} = \frac{1}{125}
\][/tex]
Therefore, the probability that Josiah will get all three questions correct by guessing is:
[tex]\[
\boxed{\frac{1}{125}}
\][/tex]
