In a survey of 60 people, it was found that 25 people read newspaper H, 26 read
newspaper T, 26 read newspaper 1, 9 read both H and I, 11 read both H and T, 8 read both T and I1, 3 read all three newspapers.
(i) the number of people who read at least one of the newspapers (ii) the number of people who read exactly one newspaper
(iii) the number of people who read newspaper H only (iv) the number of people who read newspaper T only
(v) the number of people who read newspaper I only
Answers
Given : a survey of 60 people
To Find :
(i) the number of people who read at least one of the newspapers
(ii) the number of people who read exactly one newspaper
(iii) the number of people who read newspaper H only
(iv) the number of people who read newspaper T only
(v) the number of people who read newspaper I only
Solution:
H = 25
T = 26
I = 26
H ∩ I = 9
H ∩ T = 11
T ∩ I = 8
H ∩ T ∩ I = 3
Total = 60
read at least one of the newspapers = Total - Read None
Total = H + T + I - H ∩ I - H ∩ T - T ∩ I + H ∩ T ∩ I + None
=> Total - None = 25 + 26 + 26 - 9 - 11 - 8 + 3
=> Total - None = 52
read at least one of the newspapers = 52
the number of people who read newspaper H only = H - - H ∩ I - H ∩ T + H ∩ T ∩ I
= 25 - 9 - 11 + 3 = 8
newspaper I only = 12
newspaper T only = 10
the number of people who read exactly one newspaper = 8 + 12 + 10 = 30
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SOLUTION
GIVEN
In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I , 9 read both H and I , 11 read both H and T, 8 read both T and I , 3 read all three newspapers.
TO DETERMINE
(i) The number of people who read at least one of the newspapers
(ii) The number of people who read exactly one newspaper
(iii) The number of people who read newspaper H only
(iv) The number of people who read newspaper T only
(v) The number of people who read newspaper I only
EVALUATION
Let H : The set of people who read paper H.
I : The set of people who read newspaper I
T : The set of people who read newspaper T
By the given -
n(H) = 25 , n(T) = 26 , n(I) = 26 , n(H∩I) = 9 , n(H∩T) = 11 , n(T∩I) = 8 , n(H∩T∩I) = 3
ANSWER TO QUESTION : (i)
By the formula from Set theory
n(H∪T∪T)
= n(H) + n(T) + n(I) - n(H∩T) - n(T∩I) - n(I∩H) + n(H∩T∩I)
= 25 + 26 + 26 - 9 - 11 - 8 + 3
= 52
Number of people who read at least one newspaper
= 52
ANSWER TO QUESTION : (ii)
(ii) Number of people who read exactly one paper
= n(H only) + n( T only) + n( I only )
= 8 + 10 + 12
= 30 ( From Venn diagram )
ANSWER TO QUESTION : (iii)
The number of people who read newspaper H only
= n(H only)
= 8 ( From Venn diagram )
ANSWER TO QUESTION : (iv)
The number of people who read newspaper T only
= n(T only)
= 10 ( From Venn diagram )
ANSWER TO QUESTION : (v)
The number of people who read newspaper I only
= n(I only)
= 12
Venn Diagram : Venn diagram is referred to the attachment
━━━━━━━━━━━━━━━━
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