At a certain college, 15% of the students receieved an A in their required math cousre, 10% receieved an A in their required chemistry course, and 5% receieved 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 probabilty that he receieved an A in chemistry, if he ahd an A in math?

c.) what is the probaility that he receieved an A in math, if he had an A in chemistry?

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

Draw a venn diagram.

a) P(C or M) = P(C)+P(M)-P(C and M) = .10 + .15 - .05 = 20%

b) 15% got an A in Math, 5% both, or 33.33% of those that got an A in Chem.

c) repeat logic in b)

d) if they were independent, the prob of getting A's in both would be P(C)*P(M)

In a bank call center, it is determined over a one year period that 30% of the outgoing calls are answered by a live person as opposed to being unanswered or picked up on voice mail. The center makes over 200,000 outgoing calls per year. there is an operator at the center who places 25 outgoing calls in a one hour period. What is the probability that this operator will get live answers on 7 or more calls

A blood donor center conducted a study over a one year period and found that an average of 55 donors per day checked in to the center. What is the probability that on any given day more than 75v donors will check in?

To solve these probability questions, we need to use the concepts of union, intersection, conditional probability, and independence.

Before we proceed, let's define a few events:
- A: The event that a student received an A in their math course.
- B: The event that a student received an A in their chemistry course.

Now let's answer each question step-by-step:

a.) What is the probability that he received an A in either chemistry or math?
To calculate the probability of the union of two events (A or B), we need to add the probabilities of each event and then subtract the probability of their intersection (A and B).

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

Given:
P(A) = 15%
P(B) = 10%
P(A and B) = 5%

P(A or B) = 15% + 10% - 5% = 20%

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

b.) What is the probability that he received an A in chemistry, if he had an A in math?
To find the conditional probability of event B given event A (P(B|A)), we divide the probability of the intersection of A and B by the probability of event A.

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

Given:
P(A) = 15%
P(B) = 10%
P(A and B) = 5%

P(B|A) = P(A and B) / P(A) = 5% / 15% = 1/3 ≈ 0.3333

Therefore, the probability that a student received an A in chemistry, given that they had 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?
Similarly, to find the conditional probability of event A given event B (P(A|B)), we divide the probability of the intersection of A and B by the probability of event B.

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

Given:
P(A) = 15%
P(B) = 10%
P(A and B) = 5%

P(A|B) = P(A and B) / P(B) = 5% / 10% = 1/2 = 0.5

Therefore, the probability that a student received an A in math, given that they had 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 independent if the occurrence of one event does not affect the probability of the other event. In other words, P(A|B) = P(A) and P(B|A) = P(B).

From parts b and c, we calculated:
P(A|B) = 0.5
P(B|A) = 0.3333

Since P(A|B) ≠ P(A) and P(B|A) ≠ P(B), we can conclude that the events "A in math" and "A in chemistry" are not independent.

I hope this explanation helps you understand how to solve these probability questions!