Final answer:
The probability of a student passing a 10-question true-false quiz with at least a 70% grade by guessing can be found using the binomial probability formula, calculating the probabilities of getting exactly 7, 8, 9, or 10 questions correct and summing those values.
Step-by-step explanation:
To determine the probability of a student passing a 10-question true-false quiz with a grade of at least 70 percent when guessing randomly, we need to calculate the likelihood of the student getting at least 7 out of 10 questions correct. Since each question has a 50% chance of being answered correctly (true or false), we can model this problem using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
- n is the number of trials (questions), which is 10.
- k is the number of successful trials (correct answers).
- p is the probability of success on each trial, which is 0.5.
To find the probability of getting exactly 7, 8, 9, or 10 questions correct, we need to calculate each scenario separately and then sum them up:
- P(X=7)
- P(X=8)
- P(X=9)
- P(X=10)
Finally, the sum of these probabilities will give us the probability of passing the quiz with at least 70 percent. Each of these probabilities can be found using a binomial probability table or a calculator.