Question

In a town 40% of the population read newspaper A, 30% read another newspaper B and 20% read both. A person is chosen at random from the town. The probability that the person chosen read only one paper.
(1) \[\dfrac{1}{4}\]
(2) \[\dfrac{2}{3}\]
(3) \[\dfrac{1}{3}\]
(4) \[\dfrac{3}{{10}}\]

Answer 1

Hint: Here we will assume the total population to be 100. Then we will find the number of people who read only newspapers and the number of people who read only newspaper B and then we will finally find their probability.
The probability is given by:-
\[{\text{probability}} = \dfrac{{{\text{favourable outcomes}}}}{{{\text{total outcomes}}}}\]

Complete step-by-step answer:
Let us assume that the total population is 100.
It is given that, 40% of the population read newspaper A
This implies \[n\left( A \right) = 40\]
Mow it is given that, 30% read another newspaper B
This implies \[n\left( B \right) = 30\]
Now it is given that, 20% read both A and B
This implies, \[n\left( {A \cap B} \right) = 20\]
Now we have to calculate the number of people who read only newspaper A.
Hence, we have to subtract the number of people who read both the newspapers from the total number of people who read newspaper A.
Therefore, we get:-
\[n\left( {A{\text{ only}}} \right) = n\left( A \right) - n\left( {A \cap B} \right)\]
Putting in the respective values we get:-
\[n\left( {A{\text{ only}}} \right) = 40 - 20\]
\[ \Rightarrow n\left( {A{\text{ only}}} \right) = 20\]……………………….. (1)
Now we have to calculate the number of people who read only newspaper B.
Hence, we have to subtract the number of people who read both the newspapers from the total number of people who read newspaper B.
Therefore, we get:-
\[n\left( {{\text{B only}}} \right) = n\left( B \right) - n\left( {A \cap B} \right)\]
Putting in the respective values we get:-
\[n\left( {{\text{B only}}} \right) = 30 - 20\]
\[ \Rightarrow n\left( {{\text{B only}}} \right) = 10\]………………………. (2)
Now we know that the probability is given by:-
\[{\text{probability}} = \dfrac{{{\text{favourable outcomes}}}}{{{\text{total outcomes}}}}\]
Therefore, the probability of people who read only one newspaper is given by:-
\[{\text{probability}} = \dfrac{{n\left( {{\text{A only}}} \right) + n\left( {{\text{B only}}} \right)}}{{{\text{total population}}}}\]
Putting in the respective values we get:-
\[{\text{probability}} = \dfrac{{20 + 10}}{{100}}\]
On simplification we get:-
\[{\text{probability}} = \dfrac{{30}}{{100}}\]
\[ \Rightarrow {\text{probability}} = \dfrac{3}{{10}}\]

Therefore, option (4) is correct.

Note: Students should take a note that in such questions we have to first find the number of people who read only one particular newspaper which is done by subtracting the number of people who read both the newspapers from the number of people who read one newspaper.