Final answer:
To calculate the probability of a student passing a 10-question true-false quiz with at least a 70 percent grade by guessing, add together the binomial probabilities for getting 7, 8, 9, or 10 answers correct out of 10, with each correct answer having a probability of 0.5.
Step-by-step explanation:
The probability that a student who guesses randomly on a 10-question true-false quiz will pass with at least a 70 percent grade can be calculated using the binomial probability formula. To pass with at least 70 percent, the student needs to get at least 7 out of 10 questions correct. The formula for the binomial probability of exactly k successes in n trials is: P(X = k) = C(n, k) × (p)^k × (1-p)^(n-k), where p is the probability of success on any given trial, and C(n, k) is the combination of n items taken k at a time.
Since the quiz is true-false, the probability of guessing correctly (p) is 0.5. The probabilities for getting 7, 8, 9, or 10 correct answers must be added together to find the total probability of passing:
- P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10).
Using a calculator or binomial probability table, we sum the probabilities for each of these events to get the final answer. This process involves calculating the combinations and powers for each scenario and can become complex, often requiring software or a calculator capable of computing binomial probabilities.