Question

A university offers 8 courses, numbered C1, C2, ..., C8. There are 40 students, and each takes three (different) courses, chosen at random from C1,...,C8, independent of choices of other students.

1. What is the expected number of students enrolled in course C1?
2. Are the events "Alice takes C1" and "Alice takes C2 independent? (Alice is one of the students). Prove your
answer correct.
3. What is the probability that Alice and Bob have at least one course in common? (Bob is another student.)

Answer 1

1. The expected number of students enrolled in course C1 is 5. 2. The events 'Alice takes C1' and 'Alice takes C2' are not independent. 3. The probability that Alice and Bob have at least one course in common can be found using the complementary probability approach.

Step-by-step explanation:

1. What is the expected number of students enrolled in course C1?

To find the expected number of students enrolled in course C1, we need to consider that each student takes three different courses at random. Since there are 40 students and 8 courses in total, the probability that a student chooses course C1 as one of their three courses is 1/8. Therefore, we can find the expected number of students by multiplying the number of students (40) by the probability (1/8):

Expected number of students enrolled in C1 = 40 * 1/8 = 5 students

2. Are the events 'Alice takes C1' and 'Alice takes C2' independent? (Alice is one of the students). Prove your answer correct.

To prove if two events, 'Alice takes C1' and 'Alice takes C2', are independent, we need to calculate the probability of the joint occurrence of the two events and compare it to the product of their individual probabilities.

If the events are independent, the joint probability P('Alice takes C1' AND 'Alice takes C2') should be equal to the product of their individual probabilities P('Alice takes C1') * P('Alice takes C2').

Let's assume that the probability of Alice taking C1 is p1 and the probability of Alice taking C2 is p2.

From the given information, each student takes three different courses at random, so the probability of Alice taking C1 is 1/8 and the probability of Alice taking C2 is also 1/8 (since she can choose any of the 8 courses).

Now, let's calculate the joint probability and the product of individual probabilities:

Joint probability P('Alice takes C1' AND 'Alice takes C2') = P('Alice takes C1') * P('Alice takes C2') = 1/8 * 1/8 = 1/64.

Since the joint probability is not equal to the product of individual probabilities (1/64 ≠ 1/64), we can conclude that the events 'Alice takes C1' and 'Alice takes C2' are not independent.

3. What is the probability that Alice and Bob have at least one course in common? (Bob is another student.)

To find the probability that Alice and Bob have at least one course in common, we can use the complementary probability approach. We'll calculate the probability that Alice and Bob don't have any course in common and subtract it from 1 (total probability).

The probability that Alice and Bob don't have any course in common can be calculated as follows:

P(Alice and Bob don't have any course in common) = (7/8) * (6/8) * (5/8)

This is because for the first course, Bob can choose one of the 7 courses different from Alice's course (out of the total 8 courses). For the second course, Bob can choose one of the 6 remaining courses different from Alice's first course (out of the remaining 7 courses). For the third course, Bob can choose one of the 5 remaining courses different from Alice's first two courses (out of the remaining 6 courses).

Therefore, the probability that Alice and Bob have at least one course in common is:

P(Alice and Bob have at least one course in common) = 1 - P(Alice and Bob don't have any course in common)

P(Alice and Bob have at least one course in common) = 1 - (7/8) * (6/8) * (5/8)

Answer 2

• 1. The expected number of students enrolled in course C1 is 5.
• 2. The events "Alice takes C1" and "Alice takes C2" are not independent.
• 3. The probability that Alice and Bob have at least one course in common is 23/28.

Step-by-step explanation:

1. The expected number of students enrolled in course C1 can be calculated by multiplying the probability of a student choosing C1 by the total number of students. Since there are 8 courses and each student chooses 3 courses, the probability of a student choosing C1 is 1/8. So, the expected number of students enrolled in C1 is (1/8) * 40 = 5.

2. To determine if the events "Alice takes C1" and "Alice takes C2" are independent, we can compare the probability of both events occurring together to the product of the probabilities of each event occurring separately. If the probabilities are equal, then the events are independent. The probability of Alice taking C1 is 1/8 and the probability of Alice taking C2 is also 1/8.

So, the probability of both events occurring together is (1/8) * (1/8) = 1/64.

However, the probability of Alice taking C1 and C2 together is 0, since a student can only take one course at a time. Therefore, the events are not independent.

3. To calculate the probability that Alice and Bob have at least one course in common, we can use the principle of complementary probability. We will first calculate the probability that Alice and Bob have no courses in common, and then subtract this from 1 to find the probability that they do have at least one course in common.

Total Number of Ways to Choose Courses The total number of ways for each student to choose three courses out of eight is given by the combination formula C(8,3) = 56.

Number of Ways for Alice and Bob to Have No Courses in Common For Alice and Bob to have no courses in common, we can consider the choices for each student separately. The number of ways for Alice to choose three courses out of eight is C(8,3) = 56. Once Alice has chosen her courses, there are 5 remaining courses from which Bob can choose his three courses, giving us C(5,3) = 10 ways. Therefore, the total number of ways for Alice and Bob to have no courses in common is 56 * 10 = 560.

Probability of Alice and Bob Having No Courses in Common The probability that Alice and Bob have no courses in common is given by the ratio of the number of ways for them to have no courses in common to the total number of ways they can choose their courses:

P(no common courses) = 560/56 = 10/56 = 5/28.

Probability of Alice and Bob Having at Least One Course in Common Using the principle of complementary probability, the probability that Alice and Bob have at least one course in common is given by:

P(at least one common course) = 1 - P(no common courses) = 1 - 5/28 = 23/28.