Question

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 99 students in the school. There are 38 in the Spanish class, 35 in the French class, and 17 in the German class. There are 13 students that are in both Spanish and French, 5 are in both Spanish and German, and 6 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 exactly two language classes?

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If two students are chosen randomly, what is the probability that at least one of them is taking a language class?

Leave your answers in fraction form. You can enter exact expressions such as 10*9/(23*22) if you want to avoid typing large numbers.

Answer 1

let nuber in Spanish, French and German class are A ; B and C.

N(T)=	99
N(A)=	38
N(B)=	35
N(C)=	17
N(AnB)=	13
N(BnC)=	6
N(AnC)=	5

N(AnBnC) =

2

a)exactly 2 language number of people:

N(AnB)+N(AnC)+N(BnC)-3N(AnBnC)=

18

hence   probability that he or she is taking exactly two language classes =18/99=2/11

b)

taking at least one language class

N(A)+N(B)+N(C )-N(AnB)-N(BnC)-N(AnC)+N(AnBnC)

=

68

probability that at least one of them is taking a language class=1-P(none is taking a class)

=1-(31/99)*(30/98)=1462/1617