A student takes a 10 question multiple choice quiz-  each question having 4 choices.  Suppose a student randomly picks an answer for each question.  Find the following.

Use the following Desmos Applet and a binomial distribution to answer question https://www.desmos.com/calculator/xwe9ywwjik 




A) How many ways could student get 2 out of the 10 questions correct?

Answer 1 Question 1
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B) If student got 4 questions correct, what is the percentage student got on test?

Answer 2 Question 1
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C) What is the most likely number of questions that the student will get correct on the exam?

Answer 3 Question 1
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D) What is the chance (as a percentage) that student gets exactly 2 questions correct?

Answer 4 Question 1
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E) What is the chance (as a percentage) that student gets 4 or fewer questions correct?

Answer 5 Question 1
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F) What is the chance (as a percentage) that student gets more than 4 questions correct?

Answer 6 Question 1
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G) Would it be considered unusual for student to get more than 4 questions correct on quiz? 

Answer 7 Question 1
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To answer these questions, we will use the binomial distribution model where:

- \( n = 10 \) (total number of questions),
- \( p = 0.25 \) (probability of getting a question right, since each question has 4 choices),
- \( q = 0.75 \) (probability of getting a question wrong, \( q = 1 - p \)).

For a binomial distribution, the probability of getting exactly \( k \) correct answers out of \( n \) is given by the formula:

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

where \( \binom{n}{k} \) is the binomial coefficient.

### A) How many ways could the student get 2 out of the 10 questions correct?

The number of ways to choose 2 correct answers out of 10 is given by the binomial coefficient:

\[
\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
\]

**Answer 1: 45**

### B) If the student got 4 questions correct, what is the percentage the student got on the test?

Percentage score = \( \frac{\text{correct answers}}{\text{total questions}} \times 100 \)

\[
\frac{4}{10} \times 100 = 40\%
\]

**Answer 2: 40%**

### C) What is the most likely number of questions that the student will get correct on the exam?

The most likely number of correct answers (the mode) in a binomial distribution is the number \( k \) closest to \( n \cdot p \).

\[
n \cdot p = 10 \times 0.25 = 2.5
\]

Since \( k \) must be an integer, the most likely number of correct answers is 2 or 3.

**Answer 3: 2 (or possibly 3)**

### D) What is the chance (as a percentage) that the student gets exactly 2 questions correct?

Using the binomial distribution formula:

\[
P(X = 2) = \binom{10}{2} \cdot 0.25^2 \cdot 0.75^8
\]

I'll calculate the exact value now:

\[
P(X = 2) = 45 \cdot 0.0625 \cdot 0.100112915 = 0.2804 \text{ or } 28.04\%
\]

**Answer 4: 28.04%**

### E) What is the chance (as a percentage) that the student gets 4 or fewer questions correct?

We need to calculate the cumulative probability for getting 0, 1, 2, 3, or 4 correct answers:

\[
P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
\]

I'll calculate the cumulative probability.

The sum of probabilities for \( X = 0 \) to \( X = 4 \) is approximately \( 0.967\), or **96.7%**.

**Answer 5: 96.7%**

### F) What is the chance (as a percentage) that the student gets more than 4 questions correct?

This is the complement of the previous result:

\[
P(X > 4) = 1 - P(X \leq 4) = 1 - 0.967 = 0.033
\]

So, the chance of getting more than 4 questions correct is 3.3%.

**Answer 6: 3.3%**

### G) Would it be considered unusual for the student to get more than 4 questions correct on the quiz?

Generally, an event with a probability less than 5% is considered unusual. Since the probability of getting more than 4 questions correct is 3.3%, it would indeed be considered unusual.

**Answer 7: Yes, it would be considered unusual.**

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Would you like more details on any of these answers or have any other questions?

### Related Questions:
1. What is the probability of getting exactly 5 questions correct?
2. How would the probability change if there were 5 choices per question instead of 4?
3. What is the expected number of correct answers on a 20-question quiz?
4. How does the standard deviation help in understanding the spread of correct answers?
5. What is the probability of getting at least one question correct?

**Tip:** For small numbers of trials and two outcomes (like correct/incorrect), the binomial distribution is often the best model to use.

A student takes a 10 question multiple choice quiz, each question having 4 choices. Suppose a student randomly picks an answer for each question. Find the following: A) How many ways could the student get 2 out of the 10 questions correct? B) If the student got 4 questions correct, what is the percentage the student got on the test? C) What is the most likely number of questions that the student will get correct on the exam? D) What is the chance (as a percentage) that the student gets exactly 2 questions correct? E) What is the chance (as a percentage) that the student gets 4 or fewer questions correct? F) What is the chance (as a percentage) that the student gets more than 4 questions correct? G) Would it be considered unusual for the student to get more than 4 questions correct on the quiz?

Explore how to use binomial distribution to calculate the probability of getting correct answers on a 10-question multiple choice quiz. Learn key concepts such as binomial coefficients and the probability of success. Suitable for students in Grades 9-12.

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