Question

An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. The classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 in the French class, and 16 in the German class. There are 12 students who are in both Spanish and French, 4 who are in both Spanish and German, and 6 who are in both French and German. In addition, there are 2 students taking all 3 classes. If two students are randomly chosen, what is the probability that at exactly one of them does exactly two language classes.

Answer 1

The probability that at exactly one of them does exactly two language classes is 0.32.

Explanation:

We can model this variable as a binomial random variable with sample size n=2.

The probability of success, meaning the probability that a student is in exactly two language classes can be calculated as the division between the number of students that are taking exactly two classes and the total number of students.

The number of students that are taking exactly two classes is equal to the sum of the number of students that are taking two classes, minus the number of students that are taking the three classes:

`N_2=F\&S+S\&G+F\&G-F\&S\&G=12+4+6-2=20`

Then, the probabilty of success p is:

`p=20/100=0.2`

The probability that k students are in exactly two classes can be calcualted as:

`P(x=k) = \dbinom{n}{k} p^(k)(1-p)^(n-k)\\\\\\P(x=k) = \dbinom{2}{k} 0.2^(k) 0.8^(2-k)\\\\\\`

Then, the probability that at exactly one of them does exactly two language classes is:

`P(x=1) = \dbinom{2}{1} p^(1)(1-p)^(1)=2*0.2*0.8=0.32\\\\\\`