Question

A student is taking a multiple-choice exam in which each question has two choices. Assuming that she has no knowledge of the correct answers to any of the questions, she has decided on a strategy in which she will place two balls (marked, A and B) into a box. She randomly selects one ball for each question and replaces the ball in the box. The marking on the ball will determine her answer to the question. There are six multiple-choice questions on the exam. Complete parts (a) through (d) below. a. What is the probability that she will get six questions correct? (Round to four decimal places as needed.) b. What is the probability that she will get at least five questions correct? (Round to four decimal places as needed.) c. What is the probability that she will get no questions correct? (Round to four decimal places as needed.) d. What is the probability that she will get no more than two questions correct? (Round to four decimal places as needed.)

Answer 1

The probability of getting all six questions correct is 1/64. The probability of getting at least five questions correct is 3/64. The probability of getting no questions correct is 1/64. The probability of getting no more than two questions correct is 11/32.

Step-by-step explanation:

To find the probability that the student will get six questions correct, we need to consider that there are 2 choices for each question and she will randomly select one ball for each question. So, the probability of getting one question correct is 1/2, and since there are six questions, the probability of getting all six questions correct is (1/2)^6 = 1/64.

To find the probability that the student will get at least five questions correct, we need to calculate the probability of getting five questions correct and the probability of getting six questions correct. The probability of getting five questions correct is (1/2)^5 = 1/32, and the probability of getting six questions correct is 1/64. So, the probability of getting at least five questions correct is 1/32 + 1/64 = 3/64.

To find the probability that the student will get no questions correct, we need to calculate the probability of getting every question incorrect. Since there are two choices for each question and she will randomly select one ball for each question, the probability of getting one question incorrect is also 1/2. Therefore, the probability of getting all six questions incorrect is (1/2)^6 = 1/64.

To find the probability that the student will get no more than two questions correct, we need to calculate the probabilities of getting zero, one, or two questions correct and sum them up. The probability of getting zero questions correct is (1/2)^6 = 1/64. The probability of getting one question correct is 6*(1/2)^6 = 6/64. The probability of getting two questions correct is 15*(1/2)^6 = 15/64. So, the probability of getting no more than two questions correct is 1/64 + 6/64 + 15/64 = 22/64 = 11/32.

Answer 2

The probability of getting questions correct by random guessing can be calculated using the concept of probability. The probability of getting all six questions correct is 0.0156. The probability of getting at least five questions correct is 0.03125.

Step-by-step explanation:

To solve this problem, we can use the concept of probability. Since there are two choices for each question, the probability of getting a question correct by random guessing is 1/2 (or 0.5).

a. The probability of getting all six questions correct is (0.5)⁶ = 0.0156.

b. The probability of getting at least five questions correct is the sum of the probabilities of getting five and six questions correct, which is (0.5)⁶ + (0.5)⁶ = 0.03125.

c. The probability of getting no questions correct is (0.5)⁶ = 0.0156.

d. The probability of getting no more than two questions correct is the sum of the probabilities of getting zero, one, and two questions correct, which is (0.5)⁰ + (0.5)¹ + (0.5)² = 0.875.