A student takes a 25-question multiple choice test with four options each. If
they guess at every answer, what is the probability they will guess 8 out of
the 25 correctly?

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).

A student takes a 25-question multiple choice test with four options each. If they guess at every answer, what is the probability they will guess 8 out of the 25 correctly?

Discover how to calculate the probability of guessing exactly 8 correct answers out of 25 multiple choice questions with four options each. This problem uses the binomial distribution formula, exploring the key concepts of probability. Ideal for students in Grades 10-12.

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