In Exercises 16–18, use the following information. At a college, 43% of the students are women, 15% of the students are art majors, 4% have not chosen a major, and 8% are women and art majors. A faculty member conducts an experiment and selects students at random to participate.
16. What is the probability that a randomly selected student will be either a woman or an art major?
17. What is the probability that a randomly selected student is not a woman?
18. What is the probability that a randomly selected student is either an art
major or has not chosen a major?
This can be solved by sketching a Venn diagram.
Given:
a. 43% women
b. 8% women art majors
c. 15% art major
d. 4% not chosen major
16.
From (a) and (b),
15-8=7% are male art majors
So
P(art major∪woman)
=43%+7%=50%
17.
Assuming all students are either man or woman,
P(male)=100-43=57%
18.
Art major and those who have not chosen are mutually exclusive, so we add probabilities
P(art major ∪ not chosen)
=15+4
19%
To solve these questions, we need to use the concepts of probability and set theory.
Let's define some events:
- Event A: student is a woman
- Event B: student is an art major
- Event C: student has not chosen a major
Now, let's calculate the probabilities for each event.
16. To find the probability that a randomly selected student will be either a woman or an art major, we need to calculate the union of events A and B. The formula for the probability of the union of two events is:
P(A or B) = P(A) + P(B) - P(A and B)
Here, P(A) is the probability that a randomly selected student is a woman (43%) and P(B) is the probability that a randomly selected student is an art major (15%). P(A and B) is the probability that a randomly selected student is both a woman and an art major (8%).
So, the probability of a randomly selected student being either a woman or an art major is:
P(A or B) = P(A) + P(B) - P(A and B)
= 43% + 15% - 8%
= 50%
Therefore, the probability that a randomly selected student will be either a woman or an art major is 50%.
17. To find the probability that a randomly selected student is not a woman, we can subtract the probability of event A from 100% (because the complement of A is not A).
The probability of a randomly selected student not being a woman is:
P(not A) = 100% - P(A)
= 100% - 43%
= 57%
Therefore, the probability that a randomly selected student is not a woman is 57%.
18. To find the probability that a randomly selected student is either an art major or has not chosen a major, we need to calculate the union of events B and C.
The probability of a randomly selected student being an art major is 15%, and the probability of a student having not chosen a major is 4%.
So, the probability of a randomly selected student being either an art major or having not chosen a major is:
P(B or C) = P(B) + P(C)
= 15% + 4%
= 19%
Therefore, the probability that a randomly selected student is either an art major or has not chosen a major is 19%.