A test consists of 10 true or false questions. To pass, a student must correctly answer at least eight questions. What is the probability that a student will pass the test if they guess on each question?

To solve for the probability that a student will pass by randomly guessing true or false on each question, we can use the binomial probability formula:

P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k))

where: - P(X = k) is the probability of getting exactly k successes in n trials. - "n choose k" is the binomial coefficient, calculated as n! / (k! * (n-k)!). - p is the probability of success on a single trial. - (1-p) is the probability of failure on a single trial. - n is the number of trials (questions). - k is the number of successful trials (correct answers).

Since the test consists of 10 questions, n = 10. For true or false questions, the probability of guessing correctly (p) is 1/2.

Passing the test means getting at least 8 correct answers (k ≥ 8). So, we need to calculate the probability of getting exactly 8, exactly 9, and exactly 10 correct answers and then add these probabilities together.

P(pass) = P(X = 8) + P(X = 9) + P(X = 10)

Let's start by calculating P(X = 8):

P(X = 8) = (10 choose 8) * (1/2)^8 * (1/2)^(10-8) = 45 * (1/2)^8 * (1/2)^2 = 45 * (1/2)^10

Now, calculate P(X = 9):

P(X = 9) = (10 choose 9) * (1/2)^9 * (1/2)^(10-9) = 10 * (1/2)^9 * (1/2)^1 = 10 * (1/2)^10

Finally, calculate P(X = 10):

P(X = 10) = (10 choose 10) * (1/2)^10 = 1 * (1/2)^10

Now, add these probabilities together to get the total probability of passing:

P(pass) = P(X = 8) + P(X = 9) + P(X = 10) = 45 * (1/2)^10 + 10 * (1/2)^10 + 1 * (1/2)^10 = (45 + 10 + 1) * (1/2)^10 = 56 * (1/2)^10

P(pass) = 56 / 2^10 = 56 / 1024 = 7 / 128 ≈ 0.0547

So, the probability that the student will pass the test by guessing is approximately 0.0547 or 5.47%.