A multiple-choice test consists of 10 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 she gets at most 1 of them correct?

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

(If necessary, consult a list of formulas.)

To solve this problem, we'll use the binomial probability formula because there are two possible outcomes for each question: correct or incorrect. 

### Step 1: Define the parameters
- \( n = 10 \): The number of questions.
- \( p = \frac{1}{4} = 0.25 \): The probability of getting a question correct (since there are 4 choices and only one is correct).
- \( q = 1 - p = 0.75 \): The probability of getting a question incorrect.

We are asked to find the probability that the student gets at most 1 question correct. This means we need to find \( P(X \leq 1) \), where \( X \) is the number of correct answers.

### Step 2: Use the binomial probability formula
The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
- \( k \) is the number of correct answers.

We need to calculate the probabilities for \( X = 0 \) and \( X = 1 \), and then sum them up.

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

### Step 3: Calculate the individual probabilities

**For \( X = 0 \):**
\[
P(X = 0) = \binom{10}{0} (0.25)^0 (0.75)^{10}
\]
\[
P(X = 0) = 1 \times 1 \times (0.75)^{10} \approx 0.0563
\]

**For \( X = 1 \):**
\[
P(X = 1) = \binom{10}{1} (0.25)^1 (0.75)^{9}
\]
\[
P(X = 1) = 10 \times 0.25 \times (0.75)^{9} \approx 10 \times 0.25 \times 0.1938 \approx 0.4845
\]

### Step 4: Add the probabilities
\[
P(X \leq 1) = P(X = 0) + P(X = 1) = 0.0563 + 0.4845 \approx 0.5408
\]

### Final Answer
The probability that the student gets at most 1 question correct is approximately **0.54** when rounded to two decimal places.

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the probability that the student gets exactly 2 questions correct?
2. What is the probability that the student gets at least 1 question correct?
3. How does the probability change if the test has 15 questions instead of 10?
4. How does changing the number of answer choices to 5 affect the probability?
5. What is the expected number of correct answers if the student guesses on all 10 questions?

**Tip:** When solving binomial probability problems, it’s helpful to remember that the sum of probabilities for all possible outcomes (e.g., 0 to 10 correct answers) must equal 1. This can help check your calculations.

Calculate the probability of getting at most 1 correct answer in a multiple-choice test with 10 questions, each having 4 answer choices. Detailed step-by-step solution using the binomial probability formula. Suitable for high school students studying probability and statistics.

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