Problem 2 a small community consists of three high schools...

Problem 2. A small community consists of three high schools: Bishop's School, Yeshiva School, and La Jolla School, with enrollments of 500, 300, and 200 students, respectively. On a given day, the percentage of students absent at Bishop's School, Yeshiva School, and La Jolla School are 4%, 2%, and 3%, respectively. Find the following probabilities if a student is uniformly chosen at random from the community. (a) Probability that the student is absent (b) Conditional probability that the student is from La Jolla School, given that the student is absent (c) Conditional probability that the student is from Yeshiva School, given that the student is present

Transcript

00:01 So for this problem, i've defined each one of our events just as a, b, c, and d, where event e is living in the dormitory.

00:11 Event a is the event that a randomly selected student is a freshman, and b, c, and d are sophomore, junior, and senior, respectively.

00:21 For the first question we're asked, we're asked for the probability that a randomly selected student is a junior, so event c, who does not live in the dormitory.

00:31 C and e complement.

00:34 We find that by taking 1 minus the probability of e given c, that's effectively to give us probability of e complement given c, times probability of c.

00:47 So calculating that out, that's 1 minus 0 .3 times 0 .16, for a result of 0 .112.

00:58 For the second question asked, we're looking for probability of, this is a little bit tricky here, probability of junior or senior, so c or d, who lives in a dormitory, so and e.

01:18 So that's going to be equal to probability of c and e, plus probability of d and e, which would then be equal to, and it'll go directly to the calculation here, but for c and e, that would be probability of e given c, 0 .3, times probability of c, 0 .16, plus probability of e given d, 0 .2, times probability of d, 0 .24, for a result of 0 .096.

01:55 For the third question, we're looking for the probability of a randomly student, or randomly selected student, being a freshman, given, that they live in the dormitory.

02:08 So probability of a given e, which we can calculate by taking probability of e given a times probability of a divided by probability of e.

02:20 Now we don't have probability of e explicitly.

02:24 We can find probability of e by taking the sum over all of the different values.

02:31 I'll say x is my dummy variable here, the sum of x and omega, so x takes on each one of the possible events, except for e, because we don't want to be redundant, of the probability of e given x times probability of x.

02:51 So we take probability of e given a times probability of a and so on...

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