A college has 500 students. A survey has determined that: 69 read French 37 read

Question

A college has 500 students. A survey has determined that: 69 read French 37 read German 39 read Spanislh 15 read French and German 16 read French and Spanish 7 read German and Spanish 4 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? (22/500) What is the probability a randomly selected student reads at least 1 of the 3 languages? 044 What is the probability a randomly selected student reads exactly 1 of the 3 languages? 044 What is the probability a randomly selected student does not read any of the 3 languages? 044

Explanation / Answer

a) Here we are given that:

n( F and G) = 15 and therefore n(F and G only ) = n(F and G) - n( all 3 ) = 15 - 4 = 11
n( F and S) = 16 and therefore n(F and S only ) = n(F and S) - n( all 3 ) = 16 - 4 = 12
n( S and G) = 15 and therefore n(S and G only ) = n(S and G) - n( all 3 ) = 7 - 4 = 3

Therefore the total number of students who studies exactly 2 subjects is given as:

= n(F and G only ) + n(F and S only ) + n(S and G only ) = 11 + 12 + 3 = 26

Therefore, the required probability here is computed as:

= 26/500

= 0.052

Therefore 0.052 is the required probability here.

b) We will first compute here:

n(F only ) = n(F) - n(F and G only ) - n(F and S only ) - n( all 3 ) = 69 - 11 - 12 - 4 = 42
n(G only ) = n(G) - n(F and G only ) - n(G and S only ) - n( all 3 ) = 37 - 11 - 3 - 4 = 19
n(S only ) = n(S) - n(S and G only ) - n(F and S only ) - n( all 3 ) = 39 - 3 - 12 - 4 = 20

Therefore, now total number of students who have taken at least one subject is computed as:

= 42 + 19 + 20 + 11 + 12 + 3 + 4

= 111

Therefore the required probability that students reads at least one of the three languages is computed as:

= 111/500

= 0.222

Therefore 0.222 is the required probability here.

c) Number of students who reads exactly one language

= n(F only ) + n(G only ) + n(S only )

= 42 + 19 + 20

= 81

Therefore, the required probability here is computed as:

= 81/500

= 0.162

Therefore 0.162 is the required probability here.

d) Probability that the student does not read any of the three languages

=1 - Probability that the student reads at least one of the three languages

= 1 - 0.222

= 0.778

Therefore 0.778 is the required probability here.

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