In a sample of 100 students enrolling in a university, a questionnaire indicated that 45 of them studied English, 40 studied French, 35 studied German, 20 studied both English and French, 23 studied both English and German, 19 studied both French and German, and 12 studied all three languages. Using a Venn diagram, find the probability that a randomly chosen student studied only one of the three languages.
We begin by sketching the Venn diagram as required. There were 100 students in our sample. Therefore, the sum of all the values in our Venn diagram must be 100. We will begin with the middle section, the students that studied all three languages. We are told that 12 of the 100 students studied all three languages. We are told that 19 students studied both French and German. However, 12 of these students have already been counted. And as 19 minus 12 is equal to seven, this section will contain seven students. There are seven students that studied French and German but not English.
We can repeat this process for the students that studied both English and German. 23 minus 12 is equal to 11, so there are 11 students that studied English and German but not French. As 20 minus 12 is equal to eight, there are eight students that studied English and French but not German. We have now included all the students that studied two or more languages in our Venn diagram. Next, we see that 35 students studied German in total. We already have 11, 12, and seven students inside the German section. 11 plus 12 plus seven is equal to 30, and subtracting this from 35 gives us five. This means that there were five students that only studied German.
As 40 students studied French altogether, we can calculate the number of students that only studied French by subtracting eight, 12, and seven from 40. This is equal to 13. Subtracting eight, 12, and 11 from 45, we see that 14 students studied only English. The seven values in our Venn diagram currently — 14, 11, 12, eight, 13, seven, and five — sum to 70. As 100 minus 70 is equal to 30, there must be 30 students who didn’t study any of the three languages.
We now have a completed Venn diagram. The question asks us to calculate the probability that a randomly chosen student studies only one of the three languages. From the Venn diagram, we see that 14, 13, and five students study only one of English, French, and German, respectively. 14 plus 13 plus five is equal to 32. This means that the probability of selecting a student who only studied one language is 32 out of 100. Written as a decimal, this is equal to 0.32.