Solution:

A. Using the multiplication rule to find $P(WCC)$:

• Probability of a wrong answer: $P(W) = \frac{3}{4}$ (since 3 out of 4 answers are incorrect).
• Probability of a correct answer: $P(C) = \frac{1}{4}$ (since 1 out of 4 answers is correct).
• Multiplication rule for $P(WCC)$: Multiply the probabilities for one wrong followed by two correct answers:

$P(WCC) = P(W) \cdot P(C) \cdot P(C) = \frac{3}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} = \frac{3}{64}.$

Thus, $P(WCC) = \frac{3}{64}$.

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B. Complete list of arrangements of two correct answers and one wrong answer:

The arrangements of two correct answers (C) and one wrong answer (W) are:

1. $WCC$
2. $CCW$
3. $CWC$

For each case, the probability is calculated similarly:

1. $P(WCC) = \frac{3}{64}$ (calculated above).
2. $P(CCW) = P(C) \cdot P(C) \cdot P(W) = \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{3}{4} = \frac{3}{64}.$
3. $P(CWC) = P(C) \cdot P(W) \cdot P(C) = \frac{1}{4} \cdot \frac{3}{4} \cdot \frac{1}{4} = \frac{3}{64}.$

Thus: $P(WCC) = \frac{3}{64}, \quad P(CCW) = \frac{3}{64}, \quad P(CWC) = \frac{3}{64}.$

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C. Probability of getting exactly two correct answers:

The total probability of getting exactly two correct answers is the sum of the probabilities for all three arrangements:

$P(\text{exactly two correct}) = P(WCC) + P(CCW) + P(CWC) = \frac{3}{64} + \frac{3}{64} + \frac{3}{64} = \frac{9}{64}.$

Thus, the probability is: $P(\text{exactly two correct}) = \frac{9}{64}.$

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Final Answers:

A. $P(WCC) = \frac{3}{64}$

B.

• $P(WCC) = \frac{3}{64}$
• $P(CCW) = \frac{3}{64}$
• $P(CWC) = \frac{3}{64}$

C. $P(\text{exactly two correct}) = \frac{9}{64}$

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Let me know if you'd like further clarification or detailed explanation!

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5 Related Questions:

1. How does the multiplication rule apply when more than three guesses are made?
2. What is the probability of getting no correct answers in three guesses?
3. How does the probability change if the number of answer choices increases to five?
4. What is the expected number of correct answers in this scenario?
5. How would you calculate the probability of getting at least one correct answer?

Tip:

Always verify the total probabilities add up to 1 when analyzing all possible outcomes of a probability distribution.