An elementary school is offering 3 language classes one in S

An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. These classes are open to any of the 96 students in the school. There are 41 in the Spanish class, 35 in the French class, and 24 in the German class. There are 17 students that in both Spanish and French, 7 are in both Spanish and German, and 8 are in both French and German. In addition, there are 4 students taking all 3 classes.
a) If one student is chosen randomly, what is the probability that he or she is not in any of these classes?

b)If two students are chosen randomly, what is the probability that neither of them is taking a language class?

Solution

Let

S = in spanish class
F = in french class
G = in german class

Here,

n(S) = 41
n(F) = 35
n(G) = 24
n(S n F) = 17
n(S n G) = 7
n(F n G) = 8
n(S n F n G) = 4

Thus,

n(S U F U G) = n(S) + n(F) + n(G) - n(S n F) - n(F n G) - n(S n G) + n(S n F n G) = 72

Thus,

n(not in any class) = 96 - n(S U F U G) = 96 - 72 = 24

Thus,

P(not in any class) = 24/96 = 0.25 [answer]

*****************

b)

There are 96C2 = 4560 ways to choose 2 students.

Out of the 24 not taking any classes, there are 24C2 = 276 ways to choose 2 of them.

Thus,

P(both are not taking a language class) = 276 / 4560

= 0.060526316 [answer]