To calculate the probability of guessing exactly 8 correct answers out of 25 questions on a multiple choice test where each question has 4 options, we can use the binomial probability formula. The binomial probability is given by:

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

Where:

• $n = 25$ (total number of questions),
• $k = 8$ (number of correct guesses),
• $p = \frac{1}{4}$ (probability of guessing correctly, since there are 4 options and only one correct answer).

Now, applying the values into the formula:

$P(X = 8) = \binom{25}{8} \cdot \left(\frac{1}{4}\right)^8 \cdot \left(\frac{3}{4}\right)^{17}$

First, calculate $\binom{25}{8}$, which is the binomial coefficient.

$\binom{25}{8} = \frac{25!}{8!(25-8)!}$

Let's compute this and the entire probability expression.The probability of guessing exactly 8 correct answers out of 25 questions, with each question having 4 options, is approximately 0.124, or 12.41%.

Would you like further details on this calculation or have any questions?

Here are five related questions for deeper exploration:

1. What is the probability of guessing exactly 10 correct answers out of 25?
2. How does the probability change if there were 5 options per question instead of 4?
3. What is the probability of guessing fewer than 8 correct answers?
4. How does increasing the number of questions to 30 affect the probability of guessing 8 correctly?
5. What is the expected number of correct answers when guessing on all questions?

Tip: Binomial distributions are helpful when you are dealing with a fixed number of independent trials with two possible outcomes (success or failure).