Question

Professor May B. Right often has her facts wrong, and answers each of her students' questions incorrectly with probability 1/4, independent of other questions. In each lecture, May is asked 0, 1, or 2 questions with equal probability 1/3. Let X and Y be the number of questions May is asked and the number of questions she answers wrong in a given lecture, respectively. Compute the probability of at least one wrong answer.

12/48
11/48
9/48

Answer 1

The probability that Professor May B. Right answers at least one question incorrectly during a lecture is 11/48, given the various scenarios of how many questions she is asked.

Step-by-step explanation:

The probability of Professor May B. Right giving at least one wrong answer during a lecture when each question is answered incorrectly with a probability of 1/4 can be computed by considering the various scenarios of the number of questions asked. Since May is asked 0, 1, or 2 questions with equal probability of 1/3, and answers each question incorrectly with a probability of 1/4, independent of other questions, we can calculate the probability of getting at least one question wrong as follows:

1. No questions asked: Probability of getting a question wrong is 0 (0 questions to answer wrong).
2. 1 question asked: Probability of getting it wrong is 1/4.
3. 2 questions asked: Probability of getting at least one wrong is calculated by subtracting the probability of getting both right (which is (3/4)*(3/4) = 9/16) from 1, which gives us 1 - 9/16 = 7/16.

To find the total probability, we weigh these probabilities by the chances of each number of questions being asked (1/3 for each scenario):

• (1/3)(0) + (1/3)(1/4) + (1/3)(7/16) = 0 + 1/12 + 7/48 = (4+7)/48 = 11/48.

Therefore, the probability of at least one wrong answer is 11/48.