00:01
In this problem, it is said that there are five classes taught by the same instructor.
00:07
These five classes are the classes, a, b, c, d, and e.
00:10
We're given information about how many students in each class received a positive evaluation and how many students of each class are members of a student club.
00:19
We need to find the chance that a randomly selected member of the club has failed a mid -semester test.
00:26
So let us consider the event to be f.
00:29
Let f be the event that a member of the club has failed the mid -semester test.
00:33
We need to find the probability of this.
00:36
And for that, we will be using the law of total probability.
00:45
Now, to use the law of total probability, we need a set of mutually exclusive and exhaustive events.
00:52
So let a be the event that the student is from class a, that b be the event that the student is from class b and so on.
00:58
In that case, the three events, a, b, c, d, and e will be mutually, sorry, the five events, a, b, c, d, and e, this set of five events, they will be mutually exclusive and exhaustive.
01:11
They're mutually exclusive because no student can be a member, a student of more than one class, and they are exhaustive because there are only these five classes.
01:20
There are no students other than those from these five classes who are members of the club.
01:24
So using the law of total probability, this will be clear.
01:28
F given a times p of a plus p of f given b times p of b plus p of f given c times p of c plus p of f given d plus p of f given e times p of e so let us consider p of f given a that is the probability that a student failed given that they're from class a now to set that in a 75 % of students received a positive evaluation.
02:07
So that means 100 minus 75, that's 25%.
02:11
25 % of the students in a failed.
02:15
So that means p of f given a will be 25 % of 25 by 100.
02:19
And we multiply that with p of a, which is the probability that a randomly selected member of the club is from class a.
02:25
So it has said that 12 students from a are members of the club, and we divide that by the total number of students, students in the club, that will be 12 plus 7 plus 16 plus 5 plus 6.
02:38
That is equal to 46.
02:39
So 12 by 46, that is the probability that a student is from class a...