Question

For purposes of making budget plans for staffing, a college reviewed student's year in school and area of study. Of the students, 22.5% are seniors, 25% are juniors, 25% are sophomores, and the rest are freshmen. Also, 40% of the seniors major in the area of humanities, as did 39% of the juniors, 40% of the sophomores, and 36% of the freshmen. What is the probability that a randomly selected humanities major is a junior?

Answer 1

To work a problem like this, you don't know the size of the student population, so you can't just say something like "36 humanities majors out of 100 students equals 36%".

But you do have the proportions of students in each grade overall and the percentage of those that are humanities majors.

So, a way to begin is just to say that the total number of students in the school is a variable like "x". A different way is to just assume there are 100 students total (just to make the math easy). It doesn't matter, since in the end, the actual number isn't what you need... you just need the probability.

Answer 2

So, let's use "x" students as the total population.

25% are juniors, so 0.25x of all students are juniors. If the question just asked for the probability of choosing a junior of any major, the answer would be 25%.

But you also know that only 39% of the juniors are humanities majors... in other words, 39% of 0.25x. This is the same as 0.39 * 0.25x = 0.0975x

And remember, the total number of students overall is just "x". So to get the probability of finding a junior who is a humanities major, it is:


Probability of finding a junior humanities major: P
P = # of junior humanities majors / total number of students
P = 0.0975x / x = 0.0975
P = 9.75%

As you can see, it doesn't make any difference what "x" is.

Answer 3

Oh wait, I read the problem wrong... that's not quite right... hold on please!

Answer 4

It asked for the probability of choosing a junior from a humanities major, not choosing a junior humanities major out of everyone... oops.

Let me try that again :)

Answer 5

So, you need the chance of finding a junior from a set of humanities majors.

We already found that there are 0.0975x juniors who are humanities majors, but you need the other grades as well.

Seniors: 22.5% of class & 40% humanities = 0.225 * 0.40 * x = 0.09x
Soph: 25% of class & 40% humanities = 0.25 * 0.40 * x = 0.10x
Fresh: (100-22.5-25-25) * 36% humanities
27.5% of class * 36% humanities = 0.099x

So, all humanities students added together is:
Seniors + Juniors + Soph + Freshmen
0.09x + 0.0975x + 0.10x + 0.099x = 0.3865x

Answer 6

Thank you so much!

Answer 7

And the juniors humanities majors make up 0.0975x, from earlier...

So, Probability of finding a junior out of the total humanities is:
Junior humanities major / all humanitities majors =
0.0975x / 0.3865x

= 0.25226 or about 25.2%

Answer 8

I usually don't get good replies to my questions, since their statistics.
So i was surprised...
Thank you sooo much!!!!

Answer 9

You would expect it to be about 25%, since each class has about the same percentages of of humanities majors, but the fact each class size is a little different means that it is not exactly 25%.

Answer 10

It's hard to give just a small tip or hint on one like this... but please read through the solution... it really helps to see how to work one, and you will probably be able to use similar techniques on other problems.

Glad to help :)