Question 11 Score: 0 of 4 points A student takes a 10 -question true-false quiz, but did not study and randomly guesses each answer. Find the probability that the student passes the quiz with a grade of at least \( 90 \% \) of the questions correct. Note: Remember to round the result and all the intermediate calculations to 4 decimal places. Probability \( \approx 0.0107 \)

The calculation entails determining the probability of getting at least 9 correct answers out of 10 on a true-false quiz through random guessing. Given that each question has two possible answers, the total number of outcomes is \(2^{10} = 1024\). To pass, the student must answer 9 or 10 questions correctly. Utilizing the binomial probability formula, \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( p = 0.5 \) here (for true or false), we find: - For 9 correct answers: \[ P(X=9) = \binom{10}{9} (0.5)^9 (0.5)^1 = 10 \times 0.5^{10} = 10 \times \frac{1}{1024} = 0.0098 \] - For 10 correct answers: \[ P(X=10) = \binom{10}{10} (0.5)^{10} = 1 \times \frac{1}{1024} = 0.0009765625 \] Adding these probabilities gives: \[ P(X \geq 9) \approx 0.0098 + 0.0009765625 \approx 0.0107 \] That's our final rounded probability! Engaging stat tip: You can think of it like flipping a coin! Getting 9 or 10 heads in 10 tosses seems improbable yet adds a fun twist to understanding randomness. If you love numbers, try diving deeper into discrete probability theories and explore how they apply to games and decision-making! Books like "Introduction to Probability" by Dimitri P. Bertsekas offer great insights.