Question

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.

Answer 1

Answer:

a) $P(F UR) = P(F) +P(R) -P(F and R) = 0.72+0.46-0.32=0.86$

b) $P(FUR)' = 1-P(FUR)= 1-0.86 = 0.14$

Step-by-step explanation:

Let's define the following events first:

F: The event that a course has a final exam.

R: The event that a course requires a research paper

From the info provided we have that:

$P(F) = 0.72, P(R) =0.46$ P(F and R) =0.32

So then we can create a Venn diagram as we can see on the figure attached.

a. Find the probability that a course has a final exam or a research project.

For this case we can find the probability like this:

$P(F UR) = P(F) +P(R) -P(F and R) = 0.72+0.46-0.32=0.86$

b. Find the probability that a course has NEITHER of these two requirements.

For this case we can use the complement rule and we can find the probability like this:

$P(FUR)' = 1-P(FUR)= 1-0.86 = 0.14$

And that's the same value obtained with the diagram.

Answer 2

By definition of probability:

• the probability that a course has a final exam or a research project is 86%.
• the probability that a course has NEITHER of the two requirements is 14%.

Definition of Probabitity

Probability is the greater or lesser possibility that a certain event will occur.

In other words, the probability is the possibility that a phenomenon or an event will happen, given certain circumstances. It is expressed as a percentage.

Union of events

The union of events, AUB, is the event formed by all the elements of A and B. That is, the event AUB is verified when one of the two, A or B, or both occurs. AUB is read as "A or B".

The probability of the union of two compatible events is calculated as the sum of their probabilities subtracting the probability of their intersection:

P(A∪B)= P(A) + P(B) -P(A∩B)

Complementary event

A complementary event, also called an opposite event, is made up of the inverse of the results of another event. That is, That is, given an event A, a complementary event is verified as long as the event A is not verified.

The probability of occurrence of the complementary event A' will be 1 minus the probability of occurrence of A:

P(A´)= 1- P(A)

Events and probability in this case

In first place, let's define the following events:

• F: The event that a course has a final exam.
• R: The event that a course requires a research paper

Then you know:

• P(F)= 72%= 0.72
• P(R)= 46%= 0.46
• P(F and R)= P(F∩R)= 32%= 0.32 [The intersection of events, A ∩ B, is the event formed by all the elements that are, at the same time, from A and B. That is, the event A ∩ B is verified when A and B occur simultaneously.]

In this case, considering the definition of union of eventes, the probability that a course has a final exam or a research project is calculated as:

P(F∪R)= P(F) + P(R) -P(F∩R)

P(F∪R)= 0.72 + 0.46 -0.32

P(F∪R)= 0.86= 86%

Then, the probability that a course has a final exam or a research project is 86%.

In this case, considering the definition of the complementary event and its probability, the probability that a course has NEITHER of the two requirements is calculated as:

P [(F∪R)']= 1- P(F∪R)

P [(F∪R)']= 1 - 0.86

P [(F∪R)']= 0.14= 14%

Finally, the probability that a course has NEITHER of the two requirements is 14%.

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