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

To solve this problem, we need to calculate the probability that a student randomly guesses more than 4 questions correctly on a 10-question multiple choice quiz, where each question has 4 possible choices. Since the student is guessing, the probability of guessing any single question correctly is \( \frac{1}{4} \), and the probability of guessing incorrectly is \( \frac{3}{4} \).

This situation follows a **binomial distribution** because each question has two possible outcomes: correct or incorrect. The binomial probability formula is:

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

Where:
- \( n = 10 \) (number of trials, i.e., questions),
- \( k \) is the number of correct answers (this varies),
- \( p = \frac{1}{4} \) (the probability of success, i.e., getting a question right).

We are asked to find the probability that the student gets more than 4 questions correct, i.e., \( P(X > 4) \). This is equivalent to calculating the sum of the probabilities for getting 5, 6, 7, 8, 9, and 10 correct answers:

\[
P(X > 4) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
\]

I will now calculate this probability.The probability that the student gets more than 4 questions correct by randomly guessing is approximately **7.81%**.

Would you like further details on how this was calculated or any other clarification?

Here are some related questions for practice:
1. What is the probability of getting exactly 4 questions correct?
2. What is the probability of getting all 10 questions correct?
3. What is the expected number of correct answers in this quiz?
4. How does the probability change if the quiz has 20 questions?
5. What is the variance of the number of correct answers in this scenario?

**Tip:** In binomial distributions, the expected value is calculated as \( E(X) = np \), where \( n \) is the number of trials and \( p \) is the probability of success.

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

Learn how to calculate the probability of getting more than 4 questions correct on a 10-question multiple choice quiz with 4 answer choices per question. This problem uses binomial distribution and covers key probability concepts. Suitable for students in Grades 9-12.

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