Welcome.
In this tutorial, I will be answering these probability questions using a Venn diagram.
In a class of 50 students, 30 take Philosophy, 23 take Sociology, and 12 take both.
The rectangle on the Venn diagram here represents the sample space.
That is, the collection of all possible outcomes.
Suppose this first circle represents Philosophy and second one, Sociology.
Since 12 students take both courses, we have 12 at the intersection here.
There are 30 students taking Philosophy so we have 30 minus 12 which is 18 remaining
here.
These 18 take Philosophy but not sociology.
The total in the Philosophy circle must add up to 30.
Again, there are 23 taking Sociology so we have 23 minus 12 which is 11, representing
those taking sociology only.
So, for Sociology, 12 + 11 equals 23.
Now, since we have 50 students in total, and adding these 3 values gives 41, then we have
50 minus 41 which is 9 left outside the two circles here.
In essence, the four values must add up to 50.
And we can convert them to probabilities if we want.
Or to percent.
But I’m just going to leave them as numbers.
#1.
Since 23 students take Sociology, the probability that a student takes Sociology is simply 23/50
which is 0.46. #2.
The probability of a student taking Philosophy and Sociology is the joint probability represented
by this intersection which is 12/50 or 0.24 #3.
The probability that a student does not take Philosophy can be found using the complement
rule.
That is, 1 minus the probability that a student takes Philosophy
which gives 1 minus 30/50 which gives20/50 or 0.4.
For #4, note that out of these 23 students that take Sociology, 12 of them also take
Philosophy.
So the remaining 11 here take Sociology but not Philosophy.
That is, sociology only.
Therefore, the probability is 11/50 which is 0.22.
In #5, we want the probability that a student takes Philosophy given that the student takes
Sociology.
This is basically asking, out of the students taking Sociology (that is, given Sociology),
what proportion also takes Philosophy?
Since out of the 23 students taking Sociology, 12 are also taking Philosophy,
the probability is 12/23 or 0.52.
For #6, Philosophy or Sociology means the union of the 2 events.
And that represents students that take Philosophy or Sociology or both.
We can then use the addition rule, or simply add these 3 values and divide by 50.
Adding the 3 values gives 41 so the probability is 41 divided by 50 giving 0.82.
For #7, “neither Philosophy nor Sociology” is the complement of “either Philosophy
or Sociology”.
And that refers to the 9 students outside the 2 circles here.
So the probability is 9/50 or 0.18.
Finally, for #8, the students that take Sociology or Philosophy but not both are the 18 taking
only Philosophy here and the 11 taking only Sociology here.
Adding 18 and 11 we have 29/50 for the probability, and that gives 0.58.
And that concludes this video.
Thanks for watching.