You are taking a quiz with 3 multiple-choice questions. Each question has 4 choices, and only one of which is correct. Assume that you guess the answers to all questions.

Use the multiplication rule to find the probability that the first two guesses are wrong and the third is correct. That is, find , where C denotes a correct answer and W denotes a wrong answer.
  (Please show your answer to 4 decimal places)

What is the probability of getting one correct answer and 2 wrong answers when 3 guesses are made?
Probability =  (Please show your answer to 4 decimal places)

To solve this, we'll break it down into two parts based on the multiplication rule.

### Part 1: Probability that the first two guesses are wrong and the third is correct
Let's denote the following:
- \( W \) for a wrong guess
- \( C \) for a correct guess

The probability of guessing a wrong answer (W) is:
\[
P(W) = \frac{3}{4}
\]
This is because, out of 4 choices, 3 are incorrect.

The probability of guessing a correct answer (C) is:
\[
P(C) = \frac{1}{4}
\]
This is because only 1 out of 4 choices is correct.

Now, we are asked to find the probability of getting two wrong guesses followed by one correct guess, i.e., \( P(W, W, C) \).

Using the multiplication rule, the total probability is:
\[
P(W, W, C) = P(W) \times P(W) \times P(C)
\]

Substituting the values:
\[
P(W, W, C) = \left( \frac{3}{4} \right) \times \left( \frac{3}{4} \right) \times \left( \frac{1}{4} \right)
\]
\[
P(W, W, C) = \frac{9}{64} \times \frac{1}{4} = \frac{9}{256}
\]

Converting this to decimal form:
\[
P(W, W, C) = 0.0352
\]

Thus, the probability that the first two guesses are wrong and the third is correct is:
\[
\boxed{0.0352}
\]

---

### Part 2: Probability of getting exactly one correct answer and two wrong answers

Here, we are interested in the probability of exactly one correct answer and two wrong answers. The correct answer can occur in any of the three questions, so there are three different cases:
- Case 1: The correct answer is in the first position, and the other two are wrong.
- Case 2: The correct answer is in the second position, and the other two are wrong.
- Case 3: The correct answer is in the third position, and the other two are wrong.

The probability for each of these cases is the same, and we already calculated it above:
\[
P(W, W, C) = \frac{9}{256} \quad \text{(as shown in Part 1)}
\]

Since there are 3 different cases where this can happen, the total probability is:
\[
P(\text{one correct, two wrong}) = 3 \times \frac{9}{256} = \frac{27}{256}
\]

Converting this to decimal form:
\[
P(\text{one correct, two wrong}) = 0.1055
\]

Thus, the probability of getting one correct answer and two wrong answers is:
\[
\boxed{0.1055}
\]

---

Do you need further details or have any questions?

### 5 Related Questions:
1. What is the probability of getting all 3 answers correct when guessing?
2. How does the multiplication rule apply to independent events?
3. What would the probability be if there were 5 choices per question instead of 4?
4. Can you find the probability of getting exactly 2 correct answers out of 3 guesses?
5. How would the probabilities change if partial credit was awarded for wrong answers?

### Tip:
The multiplication rule is crucial when calculating probabilities for sequences of independent events, where the outcome of one event does not affect the others.

You are taking a quiz with 3 multiple-choice questions. Each question has 4 choices, and only one of which is correct. Assume that you guess the answers to all questions. Use the multiplication rule to find the probability that the first two guesses are wrong and the third is correct. What is the probability of getting one correct answer and 2 wrong answers when 3 guesses are made? (Please show your answer to 4 decimal places)

Calculate the probability of guessing the correct and wrong answers in a 3-question multiple-choice quiz. Learn how to use the multiplication rule to find the probability of getting one correct answer and two wrong answers. Detailed step-by-step explanation included.

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