Question

In a group, every member knows at least one of the languages Hindi and Urdu, 100 members know Hindi, 50 know Urdu and 25 of them know both Hindi and Urdu. How many members are there in the group.

Answer 1

Hint: Here we will apply the formula which is $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$ , in which $P\left( {A \cup B} \right)$ represents the whole group and $P\left( A \right)$ and $P\left( B \right)$ represents specific values of the group and $P\left( {A \cap B} \right)$ represents the common values between the $P\left( A \right)$and $P\left( B \right)$.

Complete step by step solution:
Given: A group is given in which every member knows at least one of the languages Hindi and Urdu. 100 members knows Hindi, 50 members know Urdu and 25 knows both Hindi and Urdu.
We have to find here the total numbers of members in the group.
So we will apply the formula which is,
$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$ in which $P\left( {A \cup B} \right)$ represents the whole group and $P\left( A \right)$ and $P\left( B \right)$ represents specific values of the group and $P\left( {A \cap B} \right)$ represents the common values between the $P\left( A \right)$and $P\left( B \right)$.
The Hindi language is represented by $P\left( A \right)$ and Urdu language is represented by $P\left( B \right)$.
So, we have ,
$
  P\left( A \right) = 100 \\
  P\left( B \right) = 50 \\
  P\left( {A \cap B} \right) = 25 \\
$
Now, we will substitute these values in the formula $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$.
$
  P\left( {A \cup B} \right) = 100 + 50 - 25 \\
  P\left( {A \cup B} \right) = 125 \\
$
So, there are 125 total members in a group in which every member knows at least one language.


Note: The given events are not mutually exclusive events so do not think to use the formula for events of mutually exclusive, that is \[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right)\].