A multiple choice test has 10 questions. Each question has four answer choices.
a. What is the probability a student randomly guesses the answers and gets exactly six questions correct?
b. Is getting exactly 10 questions correct the same probability as getting exactly zero correct? Explain.
c. Describe the steps needed to calculate the probability of getting at least six questions correct if the student randomly guesses. You do not need to calculate this probability!
a. To calculate the probability of getting exactly six questions correct when randomly guessing on a multiple choice test with 10 questions and 4 answer choices each, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n = number of trials (10 questions)
- k = number of successes (6 questions correct)
- p = probability of success on a single trial (1/4, since there are 4 answer choices)
Plugging in the values, we get:
P(X = 6) = (10 choose 6) * (1/4)^6 * (3/4)^(10-6)
b. The probability of getting exactly 10 questions correct is not the same as getting exactly zero correct. This is because getting all questions correct requires getting all individual answers correct, while getting none correct requires getting all answers wrong. The probability of getting all answers correct is (1/4)^10, while the probability of getting all answers wrong is (3/4)^10. Since (1/4)^10 is not the same as (3/4)^10, the probabilities are not the same.
c. To calculate the probability of getting at least six questions correct when randomly guessing on a multiple choice test, you would need to find the individual probabilities of getting exactly 6, 7, 8, 9, and 10 questions correct and sum them all up. This can be done using the binomial probability formula as shown in part a, but for each individual value of k (6, 7, 8, 9, 10) and then adding up all those probabilities.