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

a.) What is the probability tha he receieved 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 receieved an A in math, if he had an A in chemistry?

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

To answer these questions, we need to create a visual representation of the problem using a Venn diagram. Let's go step by step to calculate each probability.

a.) To find the probability that a student received an A in either Chemistry or Math, we need to find the union of these two events. From the given information, we know that 15% of the students received an A in Math and 10% received an A in Chemistry, and 5% received an A in both.

To calculate the probability of either event occurring, we can use the following formula:

P(A or B) = P(A) + P(B) - P(A and B)

P(A) = 15% = 0.15
P(B) = 10% = 0.10
P(A and B) = 5% = 0.05

Substituting these values into the formula:

P(A or B) = 0.15 + 0.10 - 0.05 = 0.20

Therefore, the probability that a student received an A in either Chemistry or Math is 0.20 or 20%.

b.) To find the probability that a student received an A in Chemistry, given that they received an A in Math, we need to calculate the conditional probability.

The formula for conditional probability is:

P(A given B) = P(A and B) / P(B)

P(A and B) = 5% = 0.05
P(B) = 15% = 0.15

Substituting the values into the formula:

P(A given B) = 0.05 / 0.15 ≈ 0.33

Therefore, the probability that a student received an A in Chemistry, given they received an A in Math, is approximately 0.33 or 33%.

c.) To find the probability that a student received an A in Math, given that they received an A in Chemistry, we again need to calculate the conditional probability.

Using the same formula as in part b:

P(A given B) = P(A and B) / P(B)

P(A and B) = 5% = 0.05
P(B) = 10% = 0.10

Substituting the values into the formula:

P(A given B) = 0.05 / 0.10 = 0.5

Therefore, the probability that a student received an A in Math, given they received an A in Chemistry, is 0.5 or 50%.

d.) To determine if the events "A in Math" and "A in Chemistry" are independent, we need to compare the conditional probabilities from parts b and c with the individual probabilities from part a.

If the probabilities calculated in parts b and c match the individual probabilities from part a, then the events are independent. If they do not match, then the events are dependent.

From part a, we found that the probability of receiving an A in either subject is 20%. However, in parts b and c, we found that the conditional probabilities of receiving an A in one subject given an A in the other subject are different. Therefore, the events "A in Math" and "A in Chemistry" are dependent, not independent.

Note: In the Venn diagram, the intersection represents the overlap of students who received an A in both Math and Chemistry.