An elementary school is offering 3 language classes: one in Spanish, one in Fren

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 89 students in the school. There are 28 in the Spanish class. 29 in the French class, and 16 in the German class. There are 10 students that in both Spanish and French, 4 are in both Spanish and German, and 8 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? If two students are chosen randomly, what is the probability that neither of them is taking a language class?

Explanation / Answer

1)

: Since there are 2 students taking all 3 classes, this means that there are

• 10 2 = 8 students in Spanish and French but not German,

• 4 2 = 2 students in both Spanish and German but not French, and

• 8 2 = 4 students in both French and German but not Spanish.

Hence we have 10 + 2 + 4 = 16 students taking exactly two language classes

Required probability = 16 / 89 = 0.179

2)

Number of students taking at least one language class = Spanish French German = Spanish + French +

German - (Spanish and German but not French) - (Spanish and French but not German) - (French and German but

not French) - (Spanish and German and French)

= 28 + 29 + 16 - (4 - 2) - (10 - 2) - (8 - 2) - 2

= 55 students take at least one language class.

Therefore 89 - 55 = 34 students take no language classes.

So the probability that the 2 student is not taking any language class is 34C2/89c2

= 561/ 3916

= 0.143

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