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
[tex]1$8 play the violin, n(V)=18\\16 play the piano, n(P)=16\\ 7 play neither, n(P \cup V)' =7[/tex]
(a)The Venn diagram is attached below.
[tex]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\\[/tex]
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
[tex]P(V \cap P') = \dfrac{n(V \cap P')}{n(U)}\\= \dfrac{9}{32}\\=0.28125[/tex]
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)
[tex]= \dfrac{15}{32}\\=0.46875[/tex]
iii.Probability that the student plays the piano but not the violin
Number of Students who play piano only =17-x=17-9=8
[tex]P(V' \cap P) = \dfrac{n(V' \cap P)}{n(U)}\\= \dfrac{8}{32}\\\\=0.25[/tex]
