Question

In a survey of 120 people, it was found that 50 people read newspaper H, 52 people read newspaper T, 52 people read newspaper I, 18 read both H and I, 22 people read both H and T, 16 read both T and I, 6 read all three newspapers. Find:
i) The number of people who read exactly one newspaper.
ii) The number of people who read exactly two newspapers.
iii) The number of people who read at least one of the newspapers.

Answer 1

Hint: According to the question we have to determine (i) The number of people who read exactly one newspaper, (ii) The number of people who read exactly two newspapers, and (iii) The number of people who read at least one of the newspapers. In a survey of 120 people, it was found that 50 people read newspaper H, 52 people read newspaper T, 52 people read newspaper I, 18 read both H and I, 22 people read both H and T, 16 read both T and I, 6 read all three newspapers. So, first of all to determine the number of people who read exactly one newspaper with the help of the formula as given below,
$ \Rightarrow P(H) + P(T) + P(I) - 2(H \cap T) + P(T \cap I) + P(I \cap H) + 3P(H \cap T \cap I).........................(A)$Where, $ \cap $means it’s the intersection between the number of people who reads newspapers.
Same as, to determine the number of people who read exactly two newspapers with the help of the formula as given below,
$ \Rightarrow (H \cap T) + P(T \cap I) + P(I \cap H) - 3P(H \cap T \cap I)..................(B)$
Now, we have to find the number of people who read at least one of the newspapers with the help of the formula as given below,
$ \Rightarrow P(H \cup T \cup I) = P(H) + P(T) + P(I) - (H \cap T) + P(T \cap I) + P(H \cap T \cap I)........................(C)$
Hence, with the help of the formula above we can obtain the number of people who read at least one of the newspapers.

Complete step-by-step solution:
Given,
Number of people S = 120
Step 1: First of all we have to determine the (i) The number of people who read exactly one newspaper with the help of the formula (A) as mentioned in the solution hint. Hence, on substituting all the values in formula (A),
$
   = 50 + 52 + 52 - 2(18 + 22 + 16) + 2(6) \\
   = 154 - 2(56) + 12 \\
   = 154 - 112 + 12 \\
   = 60
 $
Step 2: Now, same as the step 1 we have to determine (ii) The number of people who read exactly two newspapers with the help of the formula (B) as mentioned in the solution hint. Hence, on substituting all the values in formula (B),
$
   = 18 + 22 + 16 - 3(6) \\
   = 56 - 18 \\
   = 38
 $
Step 3: Now, same as the step 2 we have to determine (iii) The number of people who read at least one of the newspapers with the help of the formula (C) as mentioned in the solution hint. Hence, on substituting all the values in formula (C),
$
   = 50 + 52 + 52 - (18 + 22 + 16) + 6 \\
   = 154 - 56 + 6 \\
   = 104
 $

Hence, with the help of the formula (A), (B), and (C) we have obtained (i) The number of people who read exactly one newspaper = 60, (ii) The number of people who read exactly two newspaper = 38, and (iii) The number of people who read at least one of the newspaper = 104.

Note: Mean of $ \cap $ is the intersection of two conditions or the intersection of two numbers/terms and the mean of $ \cup $ is the sum or addition of two conditions or the sum of two numbers/terms.