Three language classes are offered to 100 students: Spanish, French, and German. 28 students take Spanish, 26 take French and 16 take German. In addition 12 students take both Spanish and French, 4 take both Spanish and German, 6 take both French and German, and 2 enroll in all three.

(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?

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

Given that three language classes are offered to 100 students: Spanish, French, and German. 28 students take Spanish, 26 take French and 16 take German. In addition 12 students take both Spanish and French, 4 take both Spanish and German, 6 take both French and German, and 2 enroll in all three.

We can prepare venn diagram using the above information as in attached sheet.

In the diagram we divided the 100 students into mutually exclusive and exhaustive partitions as only spanish - 14, only French -10, only German - 8

spanish and French only - 10, French and german only -- 4, German and spanish - 2. These total to 50.

So remaining 50 are outside this i.e. who have not taken any of the three.

a) probability that they are not in any of these classes=50/100= 0.50

b) probability that they are taking exactly one language class = (14+10+8)/100= 0.32

c) Two students are independent of other.

Prob for one student in atleast one of them = 50/100 = 0.5

Hence prob for two students = 0.5*0.5 = 0.25