Problem 73 - Algebra and Trigonometry

Question

In a class of 50 students, 29 are Democrats, 11 are business majors, and 5 of the business majors are Democrats. If one student is randomly selected from the class, find the probability of choosing a. a Democrat who is not a business major. b. a student who is neither a Democrat nor a business major.

Step-by-Step Solution

Answer

a. The probability of choosing a Democrat who is not a business major is 0.48. b. The probability of choosing a student who is neither a Democrat nor a business major is 0.3.

Step 1: Identifying the sets of students

First, identify the different groups of students. There are 50 students in total. 29 students are Democrats, 11 students are business majors, and 5 students are both. This means that from the 29 Democrats, 29-5=24 are Democrats who are not business majors. From the 11 business majors, 11-5=6 are business majors who are not Democrats.

Step 2: Calculation of the probabilities for problem (a)

Then, you are asked to find the probability of selecting a Democrat who is not a business major. The probability is calculated by dividing the number of desired outcomes by the number of total outcomes. The number of desired outcomes, when a Democrat who is not a business major is selected, is 24. The total number of outcomes is 50 (the total number of students). Therefore, the probability of selecting a Democrat who is not a business major is \( \frac{24}{50} \) or 0.48.

Step 3: Calculation of the probabilities for problem (b)

For problem (b), you are asked to find the probability of selecting a student who is neither a Democrat nor a business major. You need to subtract the number of Democrats and business majors, as well as those who are both, from the total number of students. This is because you exclude those who are either Democrats, or business majors. From the total number of students (50), subtract the Democrats (29), the business majors (11), and add those who are both (5), as they have been subtracted twice. Hence, the number of students who are neither Democrats nor business majors is \(50-29-11+5 = 15\). The probability of selecting a student who is neither a Democrat nor a business major is calculated by dividing the number of desired outcomes by the number of total outcomes: \( \frac{15}{50} \) or 0.3.