Probability of Getting All Answers Correct

Let's break down the probability of answering each question correctly in this multiple-choice exam by random guessing.

Probability for One Question

Each question has 4 answer choices, and only one of them is correct. If you guess randomly, the chance of picking the correct answer for a single question is the number of correct options divided by the total number of options.

• Number of correct choices = 1
• Total number of choices = 4

So, the probability of getting one question correct by guessing is:

$$ P(\text{Correct on one question}) = \frac{\text{Number of correct choices}}{\text{Total number of choices}} = \frac{1}{4} $$

Independent Events and Total Probability

The problem states that the guesses for each question are independent. This means that the outcome of guessing one question does not affect the outcome of guessing any other question.

To find the probability of multiple independent events all happening, you multiply the probabilities of each individual event.

In this exam, there are 4 questions, and we want to find the probability of getting the correct answer for Question 1 AND Question 2 AND Question 3 AND Question 4.

The probability of getting all 4 questions correct is:

$$ P(\text{All 4 correct}) = P(\text{Q1 correct}) \times P(\text{Q2 correct}) \times P(\text{Q3 correct}) \times P(\text{Q4 correct}) $$

Since the probability of getting any single question correct is $\frac{1}{4}$, we multiply this probability by itself 4 times:

$$ P(\text{All 4 correct}) = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} $$

This can be written in a more compact form using exponents:

$$ P(\text{All 4 correct}) = \left(\frac{1}{4}\right)^4 $$

Therefore, the probability of getting all answers correct by independent random guesses for each one is $\left(\frac{1}{4}\right)^4$.