To solve this, we can model the problem as a binomial probability distribution since the student is attempting multiple independent trials (each question), with two possible outcomes (correct or incorrect) for each question. The probability of success (getting a question correct) remains constant for each question.

Problem Breakdown:

• Number of questions (trials), $n = 20$
• Probability of getting a question correct, $p = \frac{1}{4} = 0.25$ (since each question has 4 choices)
• Probability of getting a question wrong, $1 - p = 0.75$
• Number of correct answers (successes), $k = 6$

We are interested in the probability of getting exactly 6 correct answers. The binomial probability formula is:

$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}$

Where:

• $\binom{n}{k}$ is the binomial coefficient, or "n choose k," which calculates the number of ways to choose $k$ successes out of $n$ trials.
• $p^k$ is the probability of getting exactly $k$ successes.
• $(1 - p)^{n - k}$ is the probability of getting the remaining $n - k$ failures.

Step-by-Step Calculation:

1. Binomial Coefficient: $\binom{20}{6} = \frac{20!}{6!(20-6)!} = 38760$
2. Success Probability: $p^6 = (0.25)^6 = 0.00024414$
3. Failure Probability: $(1 - p)^{20-6} = (0.75)^{14} = 0.032212$

Now, plug everything into the formula:

$P(X = 6) = 38760 \cdot 0.00024414 \cdot 0.032212$

Let's calculate this.The probability that the student gets exactly 6 correct answers out of 20 multiple choice questions is approximately $0.1686$, or about 16.86%.

Would you like more details or have any questions about this?

Here are 5 related questions:

1. What is the probability of getting exactly 10 correct answers out of 20?
2. How does the probability change if each question had 5 choices instead of 4?
3. What is the expected number of correct answers for this test?
4. How would you calculate the probability of getting at least 6 correct answers?
5. What is the variance and standard deviation of correct answers in this scenario?

Tip: The binomial distribution is useful when you have independent trials with two possible outcomes and a fixed probability of success.