. At a college, 72% of courses have final exams and 46% of courses require research papers. Suppose that 32% of courses
have a research paper and a final exam. Let F be the event that a course has a final exam. Let R be the event that a course
requires a research paper.
a. Find the probability that a course has a final exam or a research project.
b. Find the probability that a course has NEITHER of these two requirements.
a. P(F ∪ R) = P(F) + P(R) - P(F ∩ R) = 0.72 + 0.46 - 0.32 = 0.86
b. P(F' ∩ R') = 1 - P(F ∪ R) = 1 - 0.86 = 0.14
a. To find the probability that a course has a final exam or a research paper, we can use the formula:
P(F or R) = P(F) + P(R) - P(F and R)
Given:
P(F) = 0.72 (72% of courses have final exams)
P(R) = 0.46 (46% of courses require research papers)
P(F and R) = 0.32 (32% of courses have both a final exam and a research paper)
Using these values, we can substitute them into the formula:
P(F or R) = 0.72 + 0.46 - 0.32
= 0.88
Therefore, the probability that a course has a final exam or a research paper is 0.88.
b. To find the probability that a course has neither a final exam nor a research paper, we can use the complement rule:
P(Not F and Not R) = 1 - P(F or R)
From part a, we know that P(F or R) = 0.88. So, substituting this value into the formula:
P(Not F and Not R) = 1 - 0.88
= 0.12
Therefore, the probability that a course has neither a final exam nor a research paper is 0.12.
To find the probability that a course has a final exam or a research project, we need to find the probability of the union of the events F and R.
a. Probability that a course has a final exam or a research project (F or R):
We can use the formula:
P(F or R) = P(F) + P(R) - P(F and R)
We are given:
P(F) = 0.72 (72% of courses have final exams)
P(R) = 0.46 (46% of courses require research papers)
P(F and R) = 0.32 (32% of courses have a research paper and a final exam)
Substituting the values into the formula:
P(F or R) = 0.72 + 0.46 - 0.32 = 0.86
Therefore, the probability that a course has a final exam or a research project is 0.86 or 86%.
b. Probability that a course has NEITHER of these two requirements:
To find the probability that a course has neither a final exam nor a research project, we can use the complement rule.
P(neither F nor R) = 1 - P(F or R)
Substituting the value of P(F or R) we found in part a:
P(neither F nor R) = 1 - 0.86 = 0.14
Therefore, the probability that a course has neither a final exam nor a research project is 0.14 or 14%.