A dean has found that 62 % of entering freshmen and 78 % of...

A dean has found that $62 \%$ of entering freshmen and $78 \%$ of community college transfers eventually graduate. Of all entering students, $73 \%$ are entering freshmen and the remainder are community college transfers. a. What is the probability that a randomly chosen entering student is an entering freshman who will eventually graduate? b. Find the probability that a randomly chosen entering student will eventually graduate. c. What is the probability that a randomly chosen entering student either is an entering freshman or will eventually graduate (or both)? d. Are the events "eventually graduates" and "enters as community college transfer" statistically independent?

A dean has found that $62 \%$ of entering freshmen and $78 \%$ of community college transfers eventually graduate. Of all entering students, $73 \%$ are entering freshmen and the remainder are community college transfers.
a. What is the probability that a randomly chosen entering student is an entering freshman who will eventually graduate?
b. Find the probability that a randomly chosen entering student will eventually graduate.
c. What is the probability that a randomly chosen entering student either is an entering freshman or will eventually graduate (or both)?
d. Are the events "eventually graduates" and "enters as community college transfer" statistically independent?

Key Concepts

Statistical Independence

Statistical independence between two events exists if the occurrence of one event does not influence the probability of the other. This is formally defined by the condition that the joint probability of both events happening is equal to the product of their individual probabilities, offering a clear criterion for assessing independence.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is essential for understanding how the likelihood of one outcome may change based on additional information, and is used to analyze scenarios where events are not independent.

Law of Total Probability

The law of total probability allows you to calculate the overall probability of an event by considering all possible mutually exclusive scenarios (or partitions) of the sample space. Each scenario's probability is weighted by the conditional probability of the event within that scenario, providing a comprehensive view of the event's likelihood.

Intersection of Events

The intersection of events refers to the scenario where two events occur simultaneously. Calculating the probability of an intersection typically involves multiplying the probability of one event by the conditional probability of the second event given that the first has occurred.

Union of Events and Inclusion-Exclusion Principle

The union of events encompasses the occurrence of at least one of multiple events, and its probability is determined using the inclusion-exclusion principle. This principle adds the probabilities of individual events and subtracts the probability of their intersection to correct for any potential overcounting that results from overlapping events.

Transcript

00:01 Well, i'm going to draw a tree diagram of the information.

00:02 I kind of read it all and said, you know, i'm going to write it this way because i see some conditional probabilities.

00:08 And we know that 73 % of entering students are freshmen, and then the remaining 27 % are community college transfer students.

00:16 And then of those who are freshmen, so this is really a conditional probability, g, given f, we know that 62 % will graduate.

00:25 So that means that 38 % will not graduate.

00:29 Maybe they'll transfer and go elsewhere.

00:31 Maybe they just won't graduate.

00:33 And here we have a higher rate.

00:35 We have 78 % are graduating.

00:37 And then we have 22 % are not graduating.

00:41 And so we have a series of five questions to answer.

00:44 So our first one is what is the probability that you randomly choose a person and it's a freshman and they graduate? and so that will be the product of these two.

00:54 That would be .73 times .62 and .73 times .62 because again remember this is a conditional probability and as is this as is this as is this so .73 times .62 gives me .4526.

01:14 Now on part b we want to find the probability that a student graduates.

01:19 Just pick a random student entering in and the probability he or she graduates and so that's going to be this probability what we just found at 0 .4526 plus this point 27 times 0 .78 so plus 0 .27 times 0 .78 and so we find out the probability of graduating is 0 .6632 so about a 66 % chance of graduating.

01:50 Now part c we want to determine what is the probability if you randomly choose a student that it is either a freshman or someone who graduates.

02:01 And so if we we could go through and say take this plus this plus this or take one minus and take these a less away, the traditional way is to take the probability of being an f which is the point seven three plus the probability of graduating which is this point six six three two and then subtracting away the intersection that we just find.

02:25 So 0 .4526...

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