Question

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

Answer 1

Answer:

(a)9

(b)I. 0.28125

II. 0.46875

III. 0.25

Step-by-step 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

$P(V \cap P') = \dfrac{n(V \cap P')}{n(U)}\\= \dfrac{9}{32}\\=0.28125$

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)

$= \dfrac{15}{32}\\=0.46875$

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

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

$P(V' \cap P) = \dfrac{n(V' \cap P)}{n(U)}\\= \dfrac{8}{32}\\\\=0.25$