A teacher has found that the probability that a student studies for a test is 0.57 , and the probability that a student gets a good grade on a test is 0.77 , and the probability that both occur is 0.47 . a. Are these events independent? b. Given that a student studies, find the probability that the student gets a good grade. c. Given that a student gets a good grade, find the probability that the student studied. a. Are these events independent? No Yes b. Given that a student studies, the probability that the student gets a good grade is (Type an integer or decimal rounded to two decimal places as needed.)

Step 1 ：Given that the probability that a student studies for a test is 0.57, the probability that a student gets a good grade on a test is 0.77, and the probability that both occur is 0.47.

Step 2 ：To determine if these events are independent, we need to check if the probability of both events (student studies and gets a good grade) is equal to the product of the probabilities of each event.

Step 3 ：If they are equal, the events are independent. If not, they are dependent.

Step 4 ：Calculating the product of the probabilities of each event, we get \(0.57 \times 0.77 = 0.439\).

Step 5 ：Since \(0.439\) is not equal to the probability of both events (\(0.47\)), the events are not independent.

Step 6 ：\(\boxed{\text{No, the events are not independent}}\).