Answer: We can use Bayes' theorem to find the probability that a randomly chosen applicant has a graduate degree, given that they are male. Bayes' theorem states:
P(A|B) = P(B|A) * P(A) / P(B)
where A and B are events, P(A|B) is the conditional probability of event A given that event B has occurred, P(B|A) is the conditional probability of event B given that event A has occurred, P(A) is the prior probability of event A, and P(B) is the prior probability of event B.
In this case, we want to find P(grad degree | male), which is the probability that an applicant has a graduate degree given that they are male. Using Bayes' theorem, we have:
P(grad degree | male) = P(male | grad degree) * P(grad degree) / P(male)
We are given that 134 applicants are male, and 40 of those males have a graduate degree. Therefore:
P(male | grad degree) = 40 / total number of applicants with a graduate degree
We are not given the total number of applicants with a graduate degree, but we can find it using the fact that 40 applicants are male and have a graduate degree, and that there are 134 male applicants in total. Therefore:
total number of applicants with a graduate degree = 40 / (40/134) = 118.33 (rounded to the nearest whole number)
Next, we need to find the prior probability of having a graduate degree, P(grad degree). We can use the total number of applicants and the number of applicants with a graduate degree to calculate this probability:
P(grad degree) = number of applicants with a graduate degree / total number of applicants = 118 / 300 = 0.3933 (rounded to four decimal places)
Finally, we need to find the prior probability of being male, P(male). This probability can be calculated similarly:
P(male) = number of male applicants / total number of applicants = 134 / 300 = 0.4467 (rounded to four decimal places)
Now, we can substitute these values into Bayes' theorem to find the probability that a randomly chosen applicant has a graduate degree, given that they are male:
P(grad degree | male) = (40 / 118) * 0.3933 / 0.4467 = 0.332 (rounded to three decimal places)
Therefore, the probability that a randomly chosen applicant has a graduate degree, given that they are male, is approximately 0.332 or 33.2%.
Explanation: