To solve this problem, we can model the situation using the binomial distribution. The reason we use this model is that each question is a Bernoulli trial (there are only two possible outcomes: correct or incorrect), and the probability of getting a correct answer is constant across all trials.

Parameters of the binomial distribution:

• $n = 21$ (the total number of questions),
• $p = \frac{1}{4}$ (the probability of answering a question correctly),
• $q = 1 - p = \frac{3}{4}$ (the probability of answering a question incorrectly),
• $X$ represents the number of correct answers, which follows a binomial distribution: $X \sim \text{Bin}(n=21, p=\frac{1}{4})$.

We are asked to find the probability that the student answers at least 7 questions correctly. In other words, we need to compute:

$P(X \geq 7) = 1 - P(X < 7) = 1 - P(X \leq 6)$

We can calculate $P(X \leq 6)$ using the binomial cumulative distribution function (CDF). This will give the probability of answering 6 or fewer questions correctly. The complement will then give the probability of answering at least 7 questions correctly.

Let me calculate this for you.The probability that the student will answer at least 7 questions correctly by guessing is approximately 0.2564, or 25.64%.

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

Here are five related questions for further exploration:

1. What is the probability of answering exactly 7 questions correctly?
2. What is the expected number of correct answers if a student guesses on all 21 questions?
3. How does the probability change if each question has 5 answer choices instead of 4?
4. What is the probability of answering at least 10 questions correctly?
5. How would the probability change if the student can eliminate one wrong answer in each question?

Tip: When dealing with binomial probability problems, the cumulative distribution function (CDF) is often a key tool for solving "at least" or "at most" type questions.