Question

Suppose that a certain college class contains 45 students. Of these, 23 are sophomores, 23 are physics majors, and 8 are neither. A student is selected at random from the class. What is the probability that the student is both a sophomore and a physics major?

(a) 1/5
(b) 1/2
(c) 2/3
(d) 3/4

Answer 1

Using the principle of inclusion-exclusion, we calculate that 9 out of 45 students are both sophomores and physics majors. Hence, the probability is 1/5.

Step-by-step explanation:

To determine the probability that a student selected from the class is both a sophomore and a physics major, we can use the principle of inclusion-exclusion: probability of Sophomores plus probability of Physics majors minus the probability of being both.

Total students = 45
Sophomores = 23
Physics majors = 23
Neither = 8
Both Sophomore and Physics Major = x

The probability of a student being a sophomore or a physics major is (23 + 23 - x)/45. Since 8 are neither, this means 45 - 8 = 37 students are either a sophomore, a physics major, or both. Therefore, 23 + 23 - x = 37. Solving for x gives us x = 9. So, 9 students are both sophomores and physics majors.

The probability that a randomly selected student is both a sophomore and a physics major is 9/45 which reduces to 1/5.