Question

In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find how many can speak Hindi only.

Answer 1

Hint: We will solve this question using the concept of sets and using the formula $P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$. Here, $P\left( A\cup B \right)$ is the total number of people, $P\left( A \right)$ is group (1) who speak Hindi and $P\left( B \right)$ is group (2) who speak English. $P\left( A\cap B \right)$ are the people who speak both English and Hindi.

Complete step-by-step answer:
It is given in the question that there are a total of 950 people in the group. Also, it is given that out of 950 people, 750 people can speak Hindi and 460 people can speak English. Now, we have to find out the number of people who speak only Hindi.
We know from the concept of sets that $\left( A\cup B \right)=\left( A \right)+\left( B \right)-\left( A\cap B \right)$.
Here, $\left( A\cup B \right)$ is the union of sets A and B and $\left( A\cap B \right)$ is the intersection of sets A and B. This can be represented in a diagram.

Applying this concept in our question, we get,
$P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$………………….. (1)
Here, $P\left( A\cup B \right)$ is the total number of people in the group given in the question. $P\left( A \right)$ is the number of people who can speak Hindi. $P\left( B \right)$ is the number of people who can speak English. $P\left( A\cap B \right)$ is the number of people who can speak both English and Hindi. So, from the question, we have,
$\begin{align}
  & P\left( A\cup B \right)=950\ people \\
 & P\left( A \right)=750\ people \\
 & P\left( B \right)=460\ people \\
\end{align}$
On putting the values of $P\left( A\cup B \right)$ as 960, $P\left( A \right)$ as 750 and $P\left( B \right)$ as 460 in equation (1), we get,
$\begin{align}
  & \Rightarrow 950=750+460-P\left( A\cap B \right) \\
 & \Rightarrow 950=1210-P\left( A\cap B \right) \\
\end{align}$
Transposing 1210 from RHS to LHS we get,
$\begin{align}
  & \Rightarrow 950=1210-P\left( A\cap B \right) \\
 & \Rightarrow -P\left( A\cap B \right)=-260 \\
\end{align}$
On multiplying (-1) in both sides, we get,
$P\left( A\cap B \right)=260$
Therefore, we can say that 260 people in the group can speak both English as well as Hindi language.
Now, we have to calculate the exact number of people who speak only Hindi. So, this can be found by subtracting the number of people who speak Hindi and the number of people who speak both Hindi and English language.
So, the number of people who speak Hindi only is given by,
$P\left( A \right)-P\left( A\cap B \right)......\left( 2 \right)$
We have, $P\left( A \right)=750$ and $P\left( A\cap B \right)=260$, so we can substitute values in equation (2), and we will get,
P required = 750 – 260 = 490.
Thus, the number of people who speak Hindi only is 490.

Note: By using the concept of sets and permutation, combination, the question of probability becomes easy to solve and the chances of error in your solution will be reduced. As in this question we have used the concept of sets $P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$ to solve this question. The mistake that can be made here is that we may find the answer as 260 and stop there. We might forget to compute the difference and write the final answer.