In a survey among 100 people, it was found that 80% of these 100 people like the TV series F.R.I.E.N.D.S and 70% of all the 100 people like the TV series G.O.T. What is the minimum number of people who like both the TV series? Note that, some people may not like any of the two series.
Answers
Solution :-
given that,
→ Total survey among = 100 people
→ People who likes Friends = 80% of 100 = 80 people = n(A)
→ People who likes GOT = 70% of 100 = 70 people = n(B)
→ People who dont like any of the two series = Let x .
then,
→ People or who likes both tv series = n(A ∩ B) = ?
we know that,
→ n(Total) = (n(A) + n (B) + x) - n(A ∩ B)
→ 100 = (80 + 70 + x) - n(A ∩ B)
→ 100 = (150 + x) - n(A ∩ B)
→ n(A ∩ B) = (150 + x) - 100
→ n(A ∩ B) = (50 + x)
now, for minimum value of n(A ∩ B) , taking x = 0, we get,
→ n(A ∩ B) = 50 + 0
→ n(A ∩ B) = 50 .
therefore, minimum number of people who like both the TV series is 50 .
Note :- since n(A ∩ B) ≤ n(A) and n(A ∩ B) ≤ n(B) , if asked maximum number of people who likes both the TV series will be 70 .
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Given : In a survey among 100 people,
it was found that 80% of these 100 people like the TV series F.R.I.E.N.D.S and 70% of all the 100 people like the TV series G.O.T.
some people may not like any of the two series.
To Find : What is the minimum number of people who like both the TV series ,
Solution:
In a survey among 100 people,
hence n(Total ) = 100
80% of these 100 people like the TV series F.R.I.E.N.D.S
=> n(F) = 80
70% of all the 100 people like the TV series G.O.T.
=> n ( G) = 70
n(F∩G) = like both series
n ( none) = x does not like any series
0 ≤ x < (100 - 80) as 80 people like TV series F.R.I.E.N.D.S
=> 0 ≤ x ≤ 20
n (Total) = n(F) + n(G) - n(F∩G) + n(none)
=> 100 = 80 + 70 - n(F∩G) + x
=> n(F∩G) = 50 + x
n(F∩G) is minimum when x is minimum = 0
and maximum when x is maximum 20
Hence min n(F∩G) = 50 + 0 = 50
minimum number of people who like both the TV series are 50
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