Expert: Josh - 3/17/2005

Question
in a group of 120 movie critics, 70 do not like Fellini, 90 like Tarkovsky, and 21 do not like both. How many critics in the group like fenelli, but not Tarkovsky?

could u please post the answer as soon as possible? thankyou.

Answer
Best to draw a Venn diagram to see these things. But i can't draw here.

Facts:
Total no. of critics, N=120.
Let F denote Fellini, T denote Tarkovssky.

In general, there are four combinations possible, these combinations are mutually exclusive.
Meaning that numbers from all four categories must add up to the total number of critics M.

We have,
category1: "F and T" (meaning like F and like T)
category2: "F and not(T)" (meaning like F and not like T)
category3: "not(F) and T" (meaning not like F and like T)
category4: "not(F) and not(T)" (meaning not like F and not like T) ...[#0]

We are told,
number disliking F,
 N(not(F))=70. ...[#1]
number liking T,
 N(T)=90 ...[#2]
number disliking both F and T,
 N(not(F),not(T))=21 ...[#3]

Taking the complement of [#1],
number liking F,
 N(F)=M-N(not(F))=120-70=50 ...[#4]
But, this is made up of people liking both F and T, people liking only F, but not T,
 N(F)=N(F,T)+N(F,not(T)) ...[#5]

Taking the complement of [#2],
number disliking T,
 N(not(T))=M-N(T)=120-90=30 ...[#6]
But, this is made up of people hating both T and F and people hating only T, but actually like F.
 N(not(T))=N(F,not(T))+N(not(F),not(T)) ...[#7]

N(F,not(T))=N(not(T))-N(not(F),not(T))
=30-21 using[#4] and [#3]
=9