At a certain college, 15% of the students received an A in their required math course, 10% receieved an A in their required chemistry course, and 5% received an A in both. A student is selected at random.

a.) What is the probability that he received an A in either chemistry or math?

b.) What is the probability that he received an A in chemistry, if he had an A in math?

c.) What is the probability that he received an A in math, if he had an A in chemistry?

d.) Are the events " A in math" and "A in chemistry" independent?

To find the probabilities in this scenario, we can use the concept of set theory and the inclusion-exclusion principle. Let's go through each question:

a.) What is the probability that he received an A in either chemistry or math?

To calculate the probability that a student received an A in either one of the courses, we need to add the individual probabilities of receiving an A in each course and subtract the probability of receiving an A in both courses.

Let's denote:
P(A_math) as the probability of receiving an A in math,
P(A_chem) as the probability of receiving an A in chemistry,
P(A_math ∩ A_chem) as the probability of receiving an A in both courses.

To find the probability that he received an A in either math or chemistry, we use the formula:
P(A_math or A_chem) = P(A_math) + P(A_chem) - P(A_math ∩ A_chem)

Given that P(A_math) = 0.15, P(A_chem) = 0.1, and P(A_math ∩ A_chem) = 0.05, we can calculate:
P(A_math or A_chem) = 0.15 + 0.1 - 0.05 = 0.2

Therefore, the probability that the student received an A in either chemistry or math is 0.2 or 20%.

b.) What is the probability that he received an A in chemistry if he had an A in math?

To find this conditional probability, we need to consider the ratio of students who received an A in both math and chemistry to the total number of students who received an A in math.

The conditional probability can be calculated using this formula:
P(A_chem | A_math) = P(A_math ∩ A_chem) / P(A_math)

Given that P(A_math ∩ A_chem) = 0.05 and P(A_math) = 0.15, we can calculate:
P(A_chem | A_math) = 0.05 / 0.15 = 1/3 = 0.3333...

Therefore, the probability that the student received an A in chemistry, given that they received an A in math, is approximately 0.3333 or 33.33%.

c.) What is the probability that he received an A in math, if he had an A in chemistry?

Similar to the previous question, we need to find the conditional probability, but this time considering the ratio of students who received an A in both math and chemistry to the total number of students who received an A in chemistry.

The conditional probability can be calculated using this formula:
P(A_math | A_chem) = P(A_math ∩ A_chem) / P(A_chem)

Given that P(A_math ∩ A_chem) = 0.05 and P(A_chem) = 0.1, we can calculate:
P(A_math | A_chem) = 0.05 / 0.1 = 1/2 = 0.5

Therefore, the probability that the student received an A in math, given that they received an A in chemistry, is 0.5 or 50%.

d.) Are the events "A in math" and "A in chemistry" independent?

Two events A and B are considered independent if the occurrence of one does not affect the probability of the other. In this case, to determine if "A in math" and "A in chemistry" are independent, we need to compare the individual probabilities with the joint probability.

If P(A) * P(B) = P(A ∩ B), then the events are independent.

Given that:
P(A_math) = 0.15,
P(A_chem) = 0.1,
P(A_math ∩ A_chem) = 0.05,

Let's calculate:
P(A_math) * P(A_chem) = 0.15 * 0.1 = 0.015,

Since P(A_math) * P(A_chem) ≠ P(A_math ∩ A_chem), i.e., 0.015 ≠ 0.05, the events "A in math" and "A in chemistry" are not independent.

Therefore, receiving an A in math and receiving an A in chemistry are dependent events.