Final answer:
The probability of getting exactly 5 questions correct on a 10-question true-false test, where the student guesses randomly on each question, is approximately 0.2461. The probability of getting 8 or more questions correct is approximately 0.0655.
Step-by-step explanation:
This is a binomial probability problem because each question can be answered correctly or incorrectly with a certain probability. Let's calculate the probability for each question.
(a) To find the probability of getting exactly 5 questions correct, we use the binomial probability formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials (10 questions), k is the number of successes (5 correct answers), C(n, k) is the number of combinations of n choose k, and p is the probability of success (0.5 because the student is guessing randomly).
Plugging in the values, we have P(X = 5) = C(10, 5) * (0.5^5) * (0.5^5) = 0.2461.
(b) To find the probability of getting 8 or more questions correct, we need to calculate the probabilities of getting 8, 9, and 10 questions correct and add them together. P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) = C(10, 8) * (0.5^8) * (0.5^2) + C(10, 9) * (0.5^9) * (0.5^1) + C(10, 10) * (0.5^10) * (0.5^0) = 0.0547 + 0.0098 + 0.00098 = 0.0655.