Question

A college has 500 students. A survey has determined that:72 read French 40 read German 39 read Spanish 18 read French and German 18 read French and Spanish 9 read German and Spanish 6 read all three languages. Put this information into a venn diagram and use it to answer the following questions. Give your answers correct to 3 decimal places. What is the probability a randomly selected student reads exactly 2 of these 3 languages?

Answer 1

Answer:

The probability that a student reads exactly 2 of these 3 languages is 0.054.

Step-by-step explanation:

Denote the languages as follows:

F = French, G = German and S = Spanish.

The information provided is:

$n(F)=72\\n(G)=40\\n(S)=39\\n(F\cap G)=18\\n(F\cap S)=18\\n(G\cap S)=9\\n(F\cap G\cap S)=6$

Compute the number of students who study only French and German as follows:

$n_{only}(F\cap G)=n(F\cap G)-(F\cap G\cap S)=18-6=12$

Compute the number of students who study only French and Spanish as follows:

$n_{only}(F\cap S)=n(F\cap S)-(F\cap G\cap S)=18-6=12$

Compute the number of students who study only German and Spanish as follows:

$n_{only}(G\cap S)=n(G\cap S)-(F\cap G\cap S)=9-6=3$

Compute the number of students who reads exactly 2 of these 3 languages as follows:

$n(2\ out\ of\ 3)=n_{only}(F\cap G)+n_{only}(F\cap S)+n_{only}(G\cap S)\\=12+12+3\\=27$

Thus, there are 27 students who reads exactly 2 of these 3 languages.

Compute the probability that a student reads exactly 2 of these 3 languages as follows:

$P(2\ out\ of\ 3)=\frac{27}{500}= 0.054$

Thus, the probability that a student reads exactly 2 of these 3 languages is 0.054.