Final answer:
The probability of a student randomly guessing on a 10-question true-false quiz and passing with at least 70 percent is found by summing the probabilities of getting 7, 8, 9, or 10 questions right. This involves using the binomial probability formula for each case and then adding the probabilities together.
Step-by-step explanation:
When a student takes a 10-question true-false quiz and guesses randomly at each answer, the probability of answering any single question correctly is 0.5. To pass the test with at least a 70 percent, the student needs to get at least 7 questions correct. We can use the binomial probability formula to find this probability, which is the sum of the probabilities of getting exactly 7, 8, 9, or 10 questions right out of 10.
P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) Where P(X = k) is the probability of getting k questions right, and it can be calculated using the formula: P(X = k) = C(n,
k) × (p)^k × (1-p)^(n-k)
Here, C(n, k) represents the binomial coefficient (combinations of n items taken k at a time), p is the probability of success on any given trial, and n is the total number of trials.
Example Calculation for P(X = 7):
C(10, 7) × (0.5)^7 × (0.5)^(10-7)
This process is repeated for k = 8, 9, and 10 and then all these probabilities are added together to get the final probability of the student passing the quiz.