Finding Conditional Probabilities In Exercises 7 and 8, use the table to find each conditional probability.
7. Business Degrees The table shows the numbers of male and female students in the United States who received bachelor's degrees in business and nonbusiness fields in a recent year. (Source: National Center for Educational Statistics)
a. Find the probability that a randomly selected bachelor's degree-earning student is male, given that the degree is in business.

Verified step by step guidance

1. Identify the conditional probability formula: P(A|B) = P(A ∩ B) / P(B). Here, we want the probability that a student is male (A), given that the degree is in business (B).
2. Determine the total number of students who earned business degrees (P(B)). From the table, this is the total in the 'Business degrees' column: 386,201.
3. Find the number of male students who earned business degrees (P(A ∩ B)). From the table, this is the value in the 'Male' row and 'Business degrees' column: 204,839.
4. Substitute the values into the conditional probability formula: P(Male | Business) = P(Male ∩ Business) / P(Business). This becomes P(Male | Business) = 204,839 / 386,201.
5. Simplify the fraction to find the conditional probability. This will give the probability that a randomly selected bachelor's degree-earning student is male, given that the degree is in business.

Key Concepts

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. It is denoted as P(A|B), which reads as the probability of event A occurring given that event B has occurred. This concept is crucial for understanding how probabilities can change based on additional information.

Joint Probability

Joint probability is the probability of two events occurring simultaneously. In the context of the question, it refers to the probability of a student being both male and earning a business degree. This concept is essential for calculating conditional probabilities, as it helps in determining the numerator in the conditional probability formula.

Total Probability

Total probability is the sum of the probabilities of all possible outcomes of a random variable. In this case, it involves understanding the total number of students earning degrees, which serves as the denominator when calculating conditional probabilities. This concept ensures that probabilities are contextualized within the entire population being studied.