Assume that random guesses are made for 8 ​multiple-choice questions on a test with 2 choices for each​ question, so that there are n=8 ​trials, each with probability of success​ (correct) given by p=.50. Find the probability of no correct answers.

This problem involves a binomial distribution, where the number of trials \( n = 8 \), the probability of success \( p = 0.50 \), and we are asked to find the probability of no correct answers. In other words, we are looking for the probability of getting zero successes, \( P(X = 0) \).

The probability mass function (PMF) of a binomial distribution is given by:

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

Where:
- \( X \) is the number of successes (correct answers),
- \( k \) is the specific number of successes (in this case, \( k = 0 \)),
- \( \binom{n}{k} \) is the binomial coefficient \( \frac{n!}{k!(n-k)!} \),
- \( p = 0.50 \) is the probability of success,
- \( n = 8 \) is the number of trials.

For \( k = 0 \), the formula simplifies to:

\[
P(X = 0) = \binom{8}{0} (0.50)^0 (0.50)^8
\]

We know that \( \binom{8}{0} = 1 \) and \( (0.50)^0 = 1 \). Thus, the probability becomes:

\[
P(X = 0) = 1 \cdot (0.50)^8 = (0.50)^8
\]

Now calculate \( (0.50)^8 \):

\[
(0.50)^8 = \frac{1}{256} = 0.00390625
\]

Therefore, the probability of no correct answers is:

\[
P(X = 0) = 0.00390625
\]

### Final Answer:
The probability of no correct answers is \( 0.0039 \) (rounded to four decimal places).

---

Do you need further details or clarifications on this solution?

Here are 5 related questions you might find interesting:
1. What is the probability of getting exactly 4 correct answers?
2. How would the probability change if there were 4 choices per question instead of 2?
3. What is the expected number of correct answers in this test?
4. How can you calculate the probability of getting at least one correct answer?
5. What is the variance of the number of correct answers in this binomial experiment?

**Tip:** In binomial distributions, if the probability of success and failure is the same, the most likely outcome is to get close to half successes in a large number of trials.

Explore how to calculate the probability of no correct answers when guessing on 8 multiple-choice questions with 2 choices each. This problem uses concepts from binomial distribution, providing a detailed step-by-step solution. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation