Use the binomial probability to answer this question. A multiple-choice test consists of 9 questions. Each question has answer choices of a, b, c, and d, and only one of the choices is correct. If a student randomly guesses on each question, what is the probability that he gets more than 1 of them correct?

Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

To solve the problem using the binomial probability formula, we need to calculate the probability that a student gets more than 1 question correct when randomly guessing on a 9-question multiple-choice test. Each question has 4 possible answers (a, b, c, d), and only one is correct.

### Given:
- The number of trials (questions) \( n = 9 \).
- The probability of success on a single trial (getting a question correct) \( p = \frac{1}{4} = 0.25 \).
- We need to find the probability that the student gets more than 1 correct answer, which is \( P(X > 1) \).

### Approach:
To find \( P(X > 1) \), we can calculate the complementary probability \( P(X \leq 1) \) and subtract it from 1.

\[
P(X > 1) = 1 - P(X \leq 1)
\]

\( P(X \leq 1) \) is the sum of \( P(X = 0) \) and \( P(X = 1) \).

### Step 1: Calculate \( P(X = 0) \)

\[
P(X = 0) = \binom{9}{0} (0.25)^0 (0.75)^9 = 1 \times 1 \times 0.75^9 = 0.1001
\]

### Step 2: Calculate \( P(X = 1) \)

\[
P(X = 1) = \binom{9}{1} (0.25)^1 (0.75)^8 = 9 \times 0.25 \times 0.75^8 = 9 \times 0.25 \times 0.1001 = 0.3003
\]

### Step 3: Calculate \( P(X \leq 1) \)

\[
P(X \leq 1) = P(X = 0) + P(X = 1) = 0.1001 + 0.3003 = 0.4004
\]

### Step 4: Calculate \( P(X > 1) \)

\[
P(X > 1) = 1 - P(X \leq 1) = 1 - 0.4004 = 0.5996
\]

### Final Answer:
Rounding the result to two decimal places, the probability that the student gets more than 1 correct answer is **0.60**.

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Would you like to dive deeper into any part of the solution, or have any additional questions?

Here are some related questions to consider:

1. What is the probability that the student gets exactly 2 correct answers?
2. How many correct answers can the student expect to get on average?
3. What is the probability that the student gets all 9 answers correct?
4. How would the probability change if there were only 5 questions?
5. What is the standard deviation of the number of correct answers?
6. How would the result change if each question had 5 possible answers instead of 4?
7. What is the probability that the student gets no questions correct?
8. How does the binomial distribution change as the number of questions increases?

**Tip:** When working with binomial probabilities, it's often easier to calculate complementary probabilities, especially when dealing with "at least" or "more than" scenarios.

Learn how to calculate binomial probability for a multiple-choice test with 9 questions. Understand the probability of getting more than 1 correct answer when guessing randomly. Detailed solution with step-by-step calculations and final answer rounded to two decimal places.

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