Question

Of the 32 students in a class, 18 play the violin, 16 play the piano, and 7 play neither.

a. Represent this in a Venn Diagram and determine how many play both the violin and
piano.

B.find the probability that the student
I.plays the violin but not the piano

ii.does not play the violin

iii.plays the piano but not the violin

Answer 1

(a)9

(b)I. 0.28125

II. 0.46875

III. 0.25

Explanation:

There are a total of 32 students, therefore the number of elements in the Universal set, n(U)=32

`1$8 play the violin, n(V)=18\\16 play the piano, n(P)=16\\ 7 play neither, n(P \cup V)' =7`

(a)The Venn diagram is attached below.

`n(U)=n(V)+n(P)-n(V \cap P)+ n(P \cup V)'\\32=18+16+7-n(V \cap P)\\32=41-n(V \cap P)\\n(V \cap P)=41-32\\n(V \cap P)=9\\`

Therefore, 9 students play both the violin and piano.

(b)

I. Probability that the student plays the violin but not the piano

Number of Students who play violin only =18-x=18-9=9

`P(V \cap P') = (n(V \cap P'))/(n(U))\\= (9)/(32)\\=0.28125`

ii.Probability that the student does not play the violin

Number of Students who does not play violin only =17-x+7=17-9+7=15

P(does not play violin only)

`= (15)/(32)\\=0.46875`

iii.Probability that the student plays the piano but not the violin

Number of Students who play piano only =17-x=17-9=8

`P(V' \cap P) = (n(V' \cap P))/(n(U))\\= (8)/(32)\\\\=0.25`