To solve this problem, we use the binomial probability distribution, as the situation involves repeated independent trials (guessing on each question) with two outcomes: correct or incorrect.

Key Information:

1. Number of trials (n): 80 questions
2. Probability of success (p): $\frac{1}{4}$ (since only one of four answers is correct)
3. Number of successes (X): We need the probability that $X$, the number of correct guesses, falls between 25 and 30 inclusive.
   • $P(25 \leq X \leq 30)$

The probability mass function for the binomial distribution is:

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

Where:

• $\binom{n}{k}$ is the binomial coefficient $\frac{n!}{k!(n-k)!}$
• $p$ is the probability of success
• $n$ is the number of trials
• $k$ is the number of successes

Since this involves a range of values, we compute:

$P(25 \leq X \leq 30) = P(X = 25) + P(X = 26) + P(X = 27) + P(X = 28) + P(X = 29) + P(X = 30)$

Exact Computation

If exact computation is required, we calculate each term $P(X = k)$ using the formula. However, this can be tedious.

Approximation with Normal Distribution

For large $n$, the binomial distribution can be approximated by the normal distribution:

$X \sim N(\mu, \sigma^2)$

Where:

• $\mu = n \cdot p = 80 \cdot \frac{1}{4} = 20$
• $\sigma = \sqrt{n \cdot p \cdot (1-p)} = \sqrt{80 \cdot \frac{1}{4} \cdot \frac{3}{4}} = \sqrt{15} \approx 3.87$

The probability $P(25 \leq X \leq 30)$ can be approximated as:

$P(25 \leq X \leq 30) \approx P\left(\frac{25 - 20}{3.87} \leq Z \leq \frac{30 - 20}{3.87}\right)$

$P(25 \leq X \leq 30) \approx P(1.29 \leq Z \leq 2.58)$

Using a standard normal distribution table:

• $P(Z \leq 2.58) \approx 0.9951$
• $P(Z \leq 1.29) \approx 0.9015$

$P(25 \leq X \leq 30) \approx 0.9951 - 0.9015 = 0.0936$

Final Answer:

The probability that the student guesses correctly on 25 to 30 questions out of 80 is approximately 0.0936 or 9.36%.

Would you like detailed steps on calculating exact binomial probabilities, or have any other questions? Here are some related questions for further exploration:

1. How is the binomial distribution derived and what are its assumptions?
2. What is the difference between exact and approximate methods for binomial probabilities?
3. How do you calculate $P(X \leq k)$ for a binomial distribution?
4. How does the normal approximation work and why is it used?
5. What factors affect the shape of the binomial distribution?

Tip: Use software tools like Python,