Questions: 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=0.50. Find the probability of no correct answers. The probability of no correct answers is

Solution Steps

We are tasked with finding the probability of getting no correct answers on a test consisting of \( n = 8 \) multiple-choice questions, each with \( p = 0.50 \) probability of answering correctly. The probability of failure (incorrect answer) is given by \( q = 1 - p = 0.50 \).

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

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

For our case, where \( x = 0 \):

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

Calculating each component:

1. The binomial coefficient \( \binom{8}{0} = 1 \).
2. The probability of success raised to the power of \( x \): \( (0.50)^0 = 1 \).
3. The probability of failure raised to the power of \( n - x \): \( (0.50)^{8} = 0.00390625 \).

Putting it all together:

\[ P(X = 0) = 1 \cdot 1 \cdot 0.00390625 = 0.00390625 \]

Rounding \( 0.00390625 \) to three decimal places gives us \( 0.004 \).

Final Answer