Questions: 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.

Solution Steps

We are tasked with finding the probability of getting no correct answers when making random guesses on 7 multiple-choice questions, each with 2 choices. This scenario can be modeled using a binomial distribution where:

• \( n = 7 \) (the number of trials),
• \( p = 0.50 \) (the probability of success, i.e., answering a question correctly),
• \( q = 1 - p = 0.50 \) (the probability of failure, i.e., answering a question incorrectly).

The probability of getting exactly \( x \) successes (correct answers) in \( n \) trials is given by the formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

For our case, we want to find the probability of getting \( x = 0 \) correct answers:

\[ P(X = 0) = \binom{7}{0} \cdot (0.50)^0 \cdot (0.50)^{7-0} \]

Calculating each component:

• The binomial coefficient \( \binom{7}{0} = 1 \),
• \( (0.50)^0 = 1 \),
• \( (0.50)^7 = \frac{1}{128} \).

Thus, we have:

\[ P(X = 0) = 1 \cdot 1 \cdot \frac{1}{128} = \frac{1}{128} \approx 0.0078125 \]

Rounding \( 0.0078125 \) to four significant digits gives us \( 0.0078 \).

Final Answer