Question

In a Biology class 60% are male and 40% are female students. It is known that in the final exam, 15% of males and 20% of the females got an A+ grade. Suppose a student is randomly selected and observed to have an A+ grade, what is the probability that the student is male?

Answer 1

The probability that a student with an A+ grade is male is approximately 52.9%. This is calculated using the given percentages and applying Bayes' theorem to determine the conditional probability.

Step-by-step explanation:

The student is asking to calculate the probability that a randomly selected student with an A+ grade is male, given the percentages of males and females in the class and the percentages of each who received an A+ grade.

Let's let M represent the event that a student is male and A+ represent the event that a student receives an A+ grade. First, we can find the probability that a student is male and receives an A+ grade (P(M ∩ A+)) and the probability that a student is female and receives an A+ grade. The initial data: P(M) = 0.60 and P(A+|M) = 0.15 for males, and P(F) = 0.40 and P(A+|F) = 0.20 for females.

Calculate the joint probabilities:
P(M ∩ A+) = P(M) × P(A+|M) = 0.60 × 0.15 = 0.09
P(F ∩ A+) = P(F) × P(A+|F) = 0.40 × 0.20 = 0.08

Next, find the probability of a student receiving an A+ grade regardless of gender:
P(A+) = P(M ∩ A+) + P(F ∩ A+) = 0.09 + 0.08 = 0.17

Finally, we calculate the probability that a student is male given that they have an A+ grade using Bayes' theorem:
P(M|A+) = P(M ∩ A+) / P(A+) = 0.09 / 0.17 ≈ 0.529

So, the probability that a student with an A+ grade is male is approximately 52.9%.