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 112 students in the school. There are 41 in the Spanish class, 37 in the French class, and 21 in the German class. There are 15 students that in both Spanish and French, 6 are in both Spanish and German, and 7 are in both French and German. In addition, there are 2 students taking all 3 classes.
If one student is chosen randomly, what is the probability that he or she is taking at least one language class?  
If two students are chosen randomly, what is the probability that at least one of them is taking a language class?

Solution

a)

Here,

n(S) = 41
n(F) = 37
n(G) = 21
n(S n F) = 15
n(S n G) = 6
n(F n G) = 7
n(S n F n G) = 2

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) = 73

Thus, the probability that a student is taking at least one class is

P(at least one class) = 73/112 = 0.651785714 [answer]

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b)

There are 112C2 = 6216 ways to choose students.

Also, note that

n(not taking a class) = 112 - n(at least one class) = 112- 73 = 39

Thus, there are 39C2 = 741 ways to choose two who don\'t take any class.

Thus,

P(both don\'t take any class) = 741/6216 = 0.119208494

Now,

P(at least one takes a class) = 1 - P(both don\'t take any class)

= 0.880791506 [ANSWER]