Question

An elementary school is offering 3 different language classes: French, Spanish and German. The classes are open to all 100 students in the school. There are 28 students in the Spanish class, 26 in French and 16 in German. There are 12 in both French and Spanish, 4 in Spanish and German and 6 in French and German. 2 are taking all 3.

1) If a student ins chosen at random, what is the probability that they are not enrolled in any language class?

2) What is the probability that the student is taking exactly 1 language class?

3) If 2 students are chosen at random, what is the probability one of

Answer 1

You have to take care to not double count people. I draw a Venn diagram with three circles for this problem. I put 2 in the very center position because 2 are taking all 3 languages. So, P(F, S, and G) = 2 / 100 = .02

Then, P(F and S, not G) = P(F and S) - P(F, S, and G) = 12/100 - 2/100 = 10/100 = .1
You continue going around the Venn diagram and collect this information:
P(F, S, and G) = 2/100
P(F, S, not G) = 10/100
P(F, G, not S) = 4 / 100
P(S, G, not F) = 2 / 100
P(S only) = 14 / 100
P(F only) = 10 / 100
P(G only) = 8 / 100

Sum all these probabilities to get P(language) = 50/100. So P(no language) = 1 - P(language) = 1 - 50 / 100 = 50/100 = .5

Answer 2

P(exactly one) = P(S only) + P(F only) + P(G only) = 32/100 = .32

Answer 3

thanks!