Question

A small town has a population of \[8000\] out of which \[3500\] people read ‘Times of India’ and\[3000\] people read ‘Indian Express’ and $800$ people read both. How many neither read both newspapers.

(A) 800

(B) 1500

(C) 2300

(D) 2850

Answer 1

Hint: Here the people who read the times of india  and indian express are given . First we must find the total number of people who read any one of the two newspapers. Later use De-Morgan's theorem to get the desired answer.

Complete step-by-step answer:

Let ‘Times of India’ be T and ‘Indian Express’ be I 

P(T) represents the probability of people reading Times of India.

P(I) represents the probability of people reading the Indian Express.

P(S) represents the probability of the total number of people who read newspapers.

As We know,

$P\left( {T \cup I} \right) = P\left( T \right) + P\left( I \right) - P\left( {T \cap I} \right) -  -  -  -  - (1)$ 

where $ \cup $ signifies union & $ \cap $ signifies intersection

Probability of total number of people read newspaper $P\left( S \right) = 8000 $

Probability of people read Times of India $P\left( T \right) = 3500 $

Probability of people read Indian Express $P\left( I \right) = 3000 $

Probability of people read both $P\left( {T \cap I} \right) = 800$

Now, we find the value of $P\left( {T \cup I} \right)$, we get,

$P\left( {T \cup I} \right) = 3500 + 3000 - 800 = 5700$

Now, by De-Morgan’s theorem, 

$P\left( {{T^c} \cap {I^c}} \right) = P{\left( {T \cup I} \right)^c}$……….. (2)

$\because $People who neither read both newspapers to be obtained, we have to find the value of $P({T^c} \cap {I^c})$

$\therefore P\left( {{T^c} \cap {I^c}} \right) = P{\left( {T \cup I} \right)^c}$

$ \Rightarrow P({T^c} \cap {I^c}) = P\left( S \right) - P\left( {T \cup I} \right)$

$ \Rightarrow P({T^c} \cap {I^c}) = 8000 - 5700 = 2300$

The number of people who neither read both newspapers is $2300$.

Hence, the correct option is (C).

Note: The concept of De-Morgan’s theorem is applied here, so we should know what it means. 

De-Morgan’s theorem states that the complement of the union of two sets is equal to the intersection of their complements and the complement of the intersection of two sets is equal to the union of their complements. 

Here, the problem can also be solved by using a Venn diagram. Venn diagram is a diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle, common elements of the sets being represented by intersections of the circle.