Probability of Getting All Answers Correct

This question asks for the probability of getting all answers correct on a multiple choice exam by random guessing. The exam has 4 questions, and each question has 4 answer choices, with only one correct answer per question.

Understanding Probability for One Question

For any single question, there are 4 possible answer choices. Since we are guessing randomly and only one choice is correct, the probability of selecting the correct answer for that single question is the number of correct choices divided by the total number of choices.

• Number of correct choices = 1
• Total number of choices = 4
• Probability of getting one question correct = $\frac{1}{4}$

Calculating Probability for Multiple Independent Questions

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

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

In this case, we want to find the probability of getting the first question correct AND the second question correct AND the third question correct AND the fourth question correct.

• Probability of getting Question 1 correct = $\frac{1}{4}$
• Probability of getting Question 2 correct = $\frac{1}{4}$
• Probability of getting Question 3 correct = $\frac{1}{4}$
• Probability of getting Question 4 correct = $\frac{1}{4}$

The probability of getting all 4 questions correct is the product of these individual probabilities:

$$ \text{Probability (all correct)} = \text{P(Q1 correct)} \times \text{P(Q2 correct)} \times \text{P(Q3 correct)} \times \text{P(Q4 correct)} $$

$$ \text{Probability (all correct)} = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} $$

$$ \text{Probability (all correct)} = \left(\frac{1}{4}\right)^4 $$

Final Probability Value

Calculating the value:

$$ \left(\frac{1}{4}\right)^4 = \frac{1^4}{4^4} = \frac{1}{256} $$

So, the probability of getting all 4 questions correct by independent random guesses is $\left(\frac{1}{4}\right)^4$ or $\frac{1}{256}$.