Questions: Ira ran out of time while taking a multiple-choice test and plans to guess on the last 6 questions. Each question has 4 possible choices, one of which is correct. Let X= the number of answers Ira correctly guesses in the last 6 questions. What is the probability that he answers fewer than 2 questions correctly in the last 6 questions? You may round your answer to the nearest hundredth.

Solution Steps

Ira is guessing on the last 6 questions of a multiple-choice test, where each question has 4 possible choices, one of which is correct. We define \( X \) as the number of questions answered correctly. We need to find the probability that he answers fewer than 2 questions correctly, i.e., \( P(X < 2) \).

To find \( P(X < 2) \), we need to calculate \( P(X = 0) \) and \( P(X = 1) \).

Using the binomial probability formula: \[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \] where:

• \( n = 6 \) (number of trials),
• \( p = \frac{1}{4} \) (probability of success),
• \( q = \frac{3}{4} \) (probability of failure).

\[ P(X = 0) = \binom{6}{0} \cdot \left(\frac{1}{4}\right)^0 \cdot \left(\frac{3}{4}\right)^6 = 1 \cdot 1 \cdot \left(\frac{729}{4096}\right) \approx 0.178 \]

\[ P(X = 1) = \binom{6}{1} \cdot \left(\frac{1}{4}\right)^1 \cdot \left(\frac{3}{4}\right)^5 = 6 \cdot \left(\frac{1}{4}\right) \cdot \left(\frac{243}{1024}\right) \approx 0.356 \]

Now, we sum the probabilities of getting fewer than 2 correct answers: \[ P(X < 2) = P(X = 0) + P(X = 1) \approx 0.178 + 0.356 = 0.534 \]

Final Answer