Answer to Question #237035 in Statistics and Probability for Ndilloh

Question

In a school of 1070 students, 530 are girls (G), 900 can swim (S) and 120 are left-handed (L). Furthermore,
there are 410 girls who can swim, 40 left-handed students who can swim, 80 girls who are left-handed and 10
left-handed girls who can swim.
a) Copy the Venn diagram below to your answer book and represent the above information in the Venn diagram
b) How many of the members neither can swim nor are left-handed?

c) Copy the Venn diagram and shade the region: (S∩G)∪(L∩S

Answer 1

a.

The box ,(Ω) represents total 1070 student

The Circle L represents total left handed student (120)

The Circle G represents total girls students (530)

The Circle S represents the students who can swim (900)

The Red part i.e intersection of Circle L and G, (L ∩ G) represents all left handed girls. (80)

The Blue part i.e intersection of Circle G and S, (G ∩ S)represents all girls who can swim (410)

The Green part i.e intersection of Circle S and L, (S  ∩ L)represents all left handed students who can swim. (40)

The Yellow part i.e inter section of Circle S, L and G, (S  ∩ L ∩ G ) represents the left handed girls who can swim (10)

b.

Note:

Ē

 is same as E^c denotes complement of a set

N (E) represents number of students in set E

The pink part represents students neither can swim i.e S^c (1070-900 = 170)

The brown part represents student who are not left handed i.e L^c ( 1070 -120 = 950)

So,

Students who neither can swim nor left handed i.e (S^c ∩ L^c) is the Grey part.

From De Morgan's Law,

N(S^c ∩ L^c) = N ( S U L)^c = N(Ω) - N (S U L)

= N(Ω) - [ N(S) + N(L) - N(S ∩ L)

= 1070 - 900 - 120 + 40

= 90

c.

Note:

(P ∩ Q)  = (Q ∩ P) 

The Blue part i.e intersection of Circle G and S, (S ∩ G) represents all girls who can swim (410)

 (L ∩ S^c) represents left handed students who cannot swim, the Sky colour part, (120 - 40 = 80)

(S ∩ G) U (L ∩ S^c) is the Orange part