Suppose Richard is taking a multiple-choice test with $10$ questions, each with $4$ answer choices, and he guesses randomly on each question. What is the probability that Richard will answer at least half of the questions correctly?

• A. $\frac{1}{4}$
• B. Less than $1\%$
• C. $\frac{1}{512}$
• D. $\frac{1}{1024}$

Answer: (B)

Verified step by step guidance

1. Identify the type of probability distribution involved. Since Richard guesses randomly on each question with 4 choices, the number of correct answers follows a Binomial distribution with parameters \(n = 10\) (number of questions) and \(p = \frac{1}{4}\) (probability of guessing correctly).
2. Define the random variable \(X\) as the number of questions Richard answers correctly. We want to find \(P(X \geq 5)\), the probability that he answers at least half (5 or more) correctly.
3. Express the probability using the binomial probability formula: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\) where \(\binom{n}{k}\) is the binomial coefficient representing the number of ways to choose \(k\) successes out of \(n\) trials.
4. Calculate the cumulative probability for \(k = 5\) to \(k = 10\): \(P(X \geq 5) = \sum_{k=5}^{10} \binom{10}{k} \left(\frac{1}{4}\right)^k \left(\frac{3}{4}\right)^{10-k}\)
5. Sum these probabilities to find the total probability that Richard answers at least half the questions correctly. This sum will give the final probability, which you can then compare to the given options.