Question

There are 3 multiple choice questions in a MCQ test. Each MCQ consists of four possible choices and only one of them is correct. If an examinee answers those MCQ randomly (without knowing the correct answers) a. What is the probability that exactly any two of the answers will be correct? b. What is the probability that at least two of the answers will be correct? c. What is the probability that at most two of the answers will be correct? d. What will be the average or expected number of correct answers? e. Also, find the standard deviation of number of correct answers

Answer 1

To answer multiple-choice questions randomly, the probabilities were calculated for different scenarios based on the likelihood of each outcome. For exactly two correct answers: 3/16. For at least two correct: 13/64. For at most two correct: 51/64. The expected number of correct answers (mean) is 3/4, and the standard deviation is √(3/16).

Step-by-step explanation:

Probability of Correct Answers in Multiple-Choice Questions

To solve the problem, we consider each multiple-choice question (MCQ) to have four possible choices with only one correct answer. There are 3 multiple-choice questions in total.

a. Probability of Exactly Two Correct Answers

The probability of getting exactly one question right is ⅓ (1/4), and incorrect is ¾ (3/4). For exactly two correct answers, there are three possible scenarios: correct-correct-incorrect, correct-incorrect-correct, or incorrect-correct-correct. The probability for each scenario is (⅓²)×(¾) = 1/16. Thus, the total probability for exactly two correct answers is 3×(1/16) = 3/16.

b. Probability of At Least Two Correct Answers

Calculating the probability of at least two correct answers means finding the probability of exactly two correct (3/16) plus the probability of all three being correct. The chance of all three being correct is (1/4)³ = 1/64. The total is 3/16 + 1/64 = 13/64.

c. Probability of At Most Two Correct Answers

The probability of at most two correct answers is the sum of probabilities of getting zero, one, or exactly two correct. Zero correct yields (3/4)³ = 27/64, one correct is 3×(1/4)×(3/4)² = 27/64, and two correct is already found as 3/16. Summing these gives 51/64.

d. Expected Number of Correct Answers

The expected number of correct answers is the sum of the products of the number of successes and their probabilities: 0×(27/64) + 1×(27/64) + 2×(3/16) + 3×(1/64) which equals 3/4.

e. Standard Deviation of the Number of Correct Answers

The standard deviation is calculated using the expected value (mean) and the probabilities of 0, 1, 2, and 3 correct answers. The variance is the sum of the squared differences between each outcome and the mean, weighted by their probabilities. Following the calculations, the standard deviation comes out to √(3/16).

Answer 2

The probability that exactly two out of three MCQs are answered correctly is 9/64; that at least two are correct is 10/64; at most two are correct is 63/64; the expected number of correct answers is 0.75; and the standard deviation is 0.46875.

Step-by-step explanation:

To solve the problems stated, let's consider that there are 3 multiple choice questions, each with 4 choices and only one correct answer. The probability of choosing the correct answer for any question is 1/4, and the probability of choosing a wrong answer is 3/4.

a. The probability that exactly two answers are correct is given by the following calculation: 3*(1/4)^2*(3/4) = 9/64 or about 0.1406, since there are three different ways this can occur (choosing the first and second, first and third, or second and third questions correctly).

b. The probability that at least two answers are correct includes the possibilities of getting exactly two or all three correct. This is calculated as P(exactly two correct) + P(all three correct) = 9/64 + (1/4)^3 = 9/64 + 1/64 = 10/64 or 0.15625.

c. The probability that at most two answers are correct includes all possibilities except getting all three correct. Thus, we have P(none correct) + P(one correct) + P(two correct) = (3/4)^3 + 3*(1/4)*(3/4)^2 + 9/64 = 27/64 + 27/64 + 9/64 = 63/64 or approximately 0.9844.

d. The expected number of correct answers can be found by the formula for the expected value of a binomial distribution, which is np, where n is the number of trials, and p is the probability of success on a single trial. For this case, it's 3*(1/4) = 3/4 or 0.75.

e. The standard deviation of the number of correct answers is given by the square root of the variance (np(1-p)). Hence, the standard deviation is sqrt(3*(1/4)*(3/4)) = sqrt(9/64) = 0.46875.