Question

A school with 70 students offers three language courses: French, Spanish, and German. 29 students in total take French, 23 students in total take Spanish, and 22 students in total take German. In addition 9 students in total are taking both French and Spanish, 7 students in total are taking both French and German, and 8 students in total are taking both Spanish and German. Finally, 2 students are taking all three courses.

(a) If a student is chosen randomly, what is the probability that they are not in any of these classes?

(b) If a student is chosen randomly, what is the probability that they are taking exactly one language class?

Answer 1

let number of students taking French, Spanish, and German are A,B and C

from above

N(T)=	70
N(A)=	29
N(B)=	23
N(C)=	22
N(AnB)=	9
N(BnC)=	8
N(AnC)=	7

N(AnBnC) =

2

a)

at least one =N(AUBUC)=	N(A)+N(B)+N(C )-N(AnB)-N(BnC)-N(AnC)+N(AnBnC)=52
none of them=N(AUBUC)c=	N(T)-N(AUBUC)=70-52=	18

b)

exactly one class =

N(A)+N(B)+N(C )-2(N(AnB)+N(BnC)+N(AnC))+3*N(AnBnC)

=32