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.