advanced probability question

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 97 students in the school. There are 40 in the Spanish class, 30 in the French class, and 21 in the German class. There are 13 students that in both Spanish and French, 4 are in both Spanish and German, and 7 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 at least one language class?
If two students are chosen randomly, what is the probability that neither of them is taking a language class?

Answer 1

Make a Venn diagram, 3 circles partially overlapping,labeled S, F, G...circle S at top left, F at top right, G somewhat below at 'middle.'

 

Place 3 where all 3 circles intersect. Place 10 above the 3, where S & F intersect--this gives total 3+10 = # who take French & Spanish or both.

 

similarly place 1 to left of 3, so that the () shaped intersection of S & G shows 4 = total who take Spanish & German or both.

also, place 4 to show that, altogether, 3+4 take French & German, or both.

 

Lastly, fill the remainder part of circle S with 26, circle F with 13, circle G with 13 so that, by adding up all 4 parts of S you get the total 40, circle F total is 30 & circle G total is 21.

 

The sum of all distinct eight areas (within circles) totals 70, I believe, leaving 27 outside any circle.

 

You now have a good logical diagram enabling you to answer questions.

 

If one student is chosen randomly, what is the probability that he or she is taking at least one language class? 70/97

 

If two students are chosen randomly, what is the probability that neither of them is taking a language class?

(20/97)(19/96)   NOTE: the last denominator should be 96...corrected after Sam's terse comment