The grades of a freshman college class, obtained after the...

The grades of a freshman college class, obtained after the first year of college, were analyzed. Seventy percent of the students in the top quarter of the college class had graduated in the upper $10 \%$ of their high school class, as had $50 \%$ of the students in the middle half of the college class and $20 \%$ of the students in the bottom quarter of the college class. a. What is the probability that a randomly chosen freshman graduated in the upper $10 \%$ of his high school class? b. What is the probability that a randomly chosen freshman who graduated in the upper $10 \%$ of the high school class will be in the top quarter of the college class? c. What is the probability that a randomly chosen freshman who did not graduate in the upper $10 \%$ of the high school class will not be in the top quarter of the college class?

The grades of a freshman college class, obtained after the first year of college, were analyzed. Seventy percent of the students in the top quarter of the college class had graduated in the upper $10 \%$ of their high school class, as had $50 \%$ of the students in the middle half of the college class and $20 \%$ of the students in the bottom quarter of the college class.
a. What is the probability that a randomly chosen freshman graduated in the upper $10 \%$ of his high school class?
b. What is the probability that a randomly chosen freshman who graduated in the upper $10 \%$ of the high school class will be in the top quarter of the college class?
c. What is the probability that a randomly chosen freshman who did not graduate in the upper $10 \%$ of the high school class will not be in the top quarter of the college class?

Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. This concept allows for the analysis of outcomes in a specific subset of a population when information about the occurrence of another event is known.

Law of Total Probability

The Law of Total Probability is a rule that relates marginal probabilities to conditional probabilities. It involves breaking down complex probability calculations by partitioning the sample space into several disjoint events, and then summing up the weighted probabilities for each partition.

Bayes' Theorem

Bayes' Theorem is a fundamental result that provides a method to update the probability estimate for an event based on new evidence. It links the conditional probability of an event given some condition with the conditional probability of the condition given the event, alongside the overall probabilities of these events.

Transcript

00:01 Well, in reading this question, i think the best thing to do is to draw a tree diagram.

00:05 And we have three categories of students.

00:08 We have those that are in the top quarter of their class.

00:14 We have those that are in the middle.

00:17 And then we have those that are in the bottom quarter.

00:22 And so naturally, if you're in the top quarter, that's 25 % of your students.

00:26 This is the middle 50%, and this is the bottom 25%.

00:31 And then we're interested in being in their high school class, being in the top 10%, and this is not.

00:41 Being in the top 10 % of the class and not, and being in their top 10 % and not.

00:53 And so we know that for this group, this is 75 % of the students, so 25 % would be here.

01:00 And this is 50 % and 50 % here.

01:04 And this is 20 % and 80 % are here.

01:09 And so we have a series of questions that we want to find.

01:12 The first one is on part a, what's the probability randomly pick a student and the person is in the upper 10%.

01:21 Well, there are three ways for that to happen.

01:25 It can be this branch, this branch, or this branch.

01:30 So we're going to take 0 .25 times .75, 75 plus 0 .5 times 0 .5 plus 0 .25 times 0 .25 times 0 .2.

01:50 And let's see what that answer comes out to be.

01:53 So 0 .25 times 0 .75 plus 0 .5 basically squared plus 0 .25 times 0 .2.

02:02 And that comes out to be 0 .4875.

02:09 Select a student, that's the likelihood they'd be in the top 10 % of the class...

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