Take a guess: A student takes a multiple-choice test that has  questions. Each question has four choices. The student guesses randomly at each answer. Let  be the number of questions answered correctly. Round the answers to at least four decimal places.

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Part 1 of 2
(a)  P (4)


This problem involves a **binomial distribution**, which is applicable when a student guesses randomly on a multiple-choice test. The parameters for a binomial distribution are:

- \( n \) = the number of trials (questions)
- \( p \) = the probability of success on each trial (answering a question correctly)
- \( X \) = the number of questions answered correctly

For this case:

- Each question has 4 choices, so the probability of guessing correctly is \( p = \frac{1}{4} = 0.25 \).
- The number of trials \( n \) is the total number of questions.
- We are asked to find \( P(X = 4) \), the probability that the student answers exactly 4 questions correctly.

The probability mass function for a binomial distribution is:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes (correct answers) out of \( n \) trials (questions).
- \( p \) is the probability of success.
- \( k \) is the number of correct answers we are calculating the probability for.

To provide a concrete answer, I need the total number of questions \( n \). Could you provide that detail?

A student takes a multiple-choice test that has questions. Each question has four choices. The student guesses randomly at each answer. Let X be the number of questions answered correctly. Round the answers to at least four decimal places.

Part 1 of 2
(a) P(X = 4)

Learn how to calculate the probability of answering 4 questions correctly on a multiple-choice test where a student guesses randomly. This problem involves binomial distribution and is suitable for students in Grades 10-12. Explore key probability concepts and formulas with detailed explanations.

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