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 107 students in the school. There are 42 in the Spanish class, 32 in the French class, and 23 in the German class. There are 14 students that in both Spanish and French, 8 are in both Spanish and German, and 9 are in both French and German. In addition, there are 4 students taking all 3 classes.

Question

If one student is chosen randomly, what is the probability that he or she is taking at least two language classes?
If two students are chosen randomly, what is the probability that at least one of them is taking German?

Answer 1

Make a Venn diagram, and you can easily see that there are 70 students taking at least one of the classes, and thus 37 taking no class.

Using the diagram, you can easily see how many students fit the questions; and thus the probabilities needed.