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To find the probability of getting no correct answers when making random guesses on a test with 7 multiple-choice questions, we can use the binomial distribution.

Given:
- \( n = 7 \) (number of trials or questions)
- \( p = 0.5 \) (probability of success, i.e., getting a question correct)

The binomial probability formula is:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Where:
- \( \binom{n}{k} \) is the binomial coefficient
- \( k \) is the number of successes (in this case, correct answers)

For no correct answers (\( k = 0 \)):
\[ P(X = 0) = \binom{7}{0} (0.5)^0 (1-0.5)^7 \]

Calculate the binomial coefficient:
\[ \binom{7}{0} = 1 \]

Now, substitute the values:
\[ P(X = 0) = 1 \cdot (0.5)^0 \cdot (0.5)^7 \]
\[ P(X = 0) = 1 \cdot 1 \cdot (0.5)^7 \]
\[ P(X = 0) = (0.5)^7 \]
\[ P(X = 0) = \frac{1}{2^7} \]
\[ P(X = 0) = \frac{1}{128} \]

Therefore, the probability of getting no correct answers is:
\[ \boxed{\frac{1}{128}} \approx 0.0078125 \]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the probability of getting exactly one correct answer?
2. What is the probability of getting exactly two correct answers?
3. What is the expected number of correct answers when making random guesses?
4. How does the probability change if the number of choices per question increases to 4?
5. What is the probability of getting at least one correct answer?

**Tip:** The sum of all probabilities in a binomial distribution for \( n \) trials is always equal to 1.

Assume that random guesses are made for 7 multiple-choice questions on a test with 2 choices for each question, so that there are n = 7 trials, each with probability of success (correct) given by p = 0.50. Find the probability of no correct answers.

Learn how to calculate the probability of getting no correct answers in a test of 7 multiple-choice questions with random guesses. This problem covers the key concepts of binomial distribution and uses the binomial probability formula. Suitable for students in Grades 10-12.

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