Question

. Every Tuesday afternoon during the school year, a certain university brought in a visiting speaker to present a lecture on some topic of current interest. On the day after the fourth lecture of the year, a random sample 250 students was selected from the student body at the university, and each of these students was asked how many of the four lectures they had attended. The counts for each combination of number of lectures and classification are given in the table below. number of lectures attended 0 1 2 3 4 freshman 14 19 20 4 13 classification sophomore 10 16 27 6 11 junior 15 15 17 4 9 senior 19 8 6 5 12 Suppose that an student is selected at random from this group. Let A denote the event that the selected student is a freshman, B denote the event that the selected student attended 3 lectures.

a) Calculate P(A), P(B), and P(A ∩ B).

b) Calculate both P(A | B) and P(B | A), and explain, in context, what each of these probabilities represents.

c) Calculate the probability that the selected student attended at least 2 lectures.

d) If the selected student attended at least 2 lectures, what is the probability that he or she is a junior?

Answer 1

number of lectures attended	0	1	2	3	4	Totals
Freshman	14	19	20	4	13	70
Sophomore	10	16	27	6	11	70
Junior	15	15	17	4	9	60
Senior	19	8	6	5	12	50
Totals	58	58	70	19	45

(a) From the above table,

P(A) = P (selected student is a freshman) = 70/250 = 0.28

P(B) = P(the selected student attended 3 lectures) = 19/250 = 0.076

(b) P (A|B) = Probability that the selected student is a freshman, given that the selected student attended 3 lectures.

There are 19 students who attended 3 lectures. There are are 4 freshman in this pool of 19 students.

Hence, probability that a selected student is a freshman, given that he has attended 3 lectures = 4/19 = 0.211

P (B|A) = Probability that the selected student has attended 3 lectures, given that he is a freshman.

There are 70 freshman in the group. Out of them, there are 4 freshman who attended 3 lectures.

Hence, probability that a selected student has attended 3 lectures, given that he is a freshman = 4/70 = 0.057

(c) Number of students who attended 2 or more lectures = 58+70+19+45 = 192

P (a randomly selected student attended 2 or more lectures) = 192/250 = 0.768

(d) Out of the 192 students who attended 2 or more lectures, no.of students who are juniors = 15+17+4+9= 45

P (a student is a junior, given that he has attended 2 or more lectures) = 45 / 192 = 0.2344