Question

A survey conducted on 600 students of B.A part I classes of a college gave the following report. Out of 600 students, 307 took economics, 198 took history, 230 took sociology, 65 took history and economics, 45 took economics and sociology, 31 took sociology and history and 10 took all the three subjects. The report sounded very impressive, but the surveyor was fired. Why?

Answer 1

Hint: The total number of students are given as 600. So, first we will find the probability of students for all the subjects using the formula, $P=\dfrac{\text{favourable outcome of an event}}{\text{total number of outcomes}}$. Then we will find the probability of the students taking one or more subjects, that is, $P\left( \text{at least 1 or more subjects} \right)=P\left( E\cup H\cup S \right)$, where E represents Economics, H represents History and S represents Sociology. We will use the formula to find it as, $P\left( E\cup H\cup S \right)=P\left( E \right)+P\left( H \right)+P\left( S \right)-P\left( E\cap H \right)-P\left( H\cap S \right)-P\left( E\cap S \right)+P\left( E\cap H\cap S \right)$

Complete step by step solution:

We are given that there a total of 600 students. And out of that 307 students took Economics, so the probability of students taking economics is given by the formula, $P=\dfrac{\text{favourable outcome of an event}}{\text{total number of outcomes}}$. So, we can say,
$P\left( E \right)=\dfrac{307}{600}\ldots \ldots \ldots \left( i \right)$
Similarly, we have been given that 198 students took History, so we get,
$P\left( H \right)=\dfrac{198}{600}\ldots \ldots \ldots \left( ii \right)$
And, 230 students took Sociology, so we get,
$P\left( S \right)=\dfrac{230}{600}\ldots \ldots \ldots \left( iii \right)$
Now, there were 65 students who took History and Economics. So, we get,
$P\left( E\cap H \right)=\dfrac{65}{600}\ldots \ldots \ldots \left( iv \right)$
And 45 students took Economics and Sociology. So, we have,
$P\left( E\cap S \right)=\dfrac{45}{600}\ldots \ldots \ldots \left( v \right)$
And 31 of them took History and Sociology. So, we get,
$P\left( H\cap S \right)=\dfrac{31}{600}\ldots \ldots \ldots \left( vi \right)$
And lastly, there were 10 students who took History, Economics and Sociology. So, we get,
$P\left( E\cap H\cap S \right)=\dfrac{10}{600}\ldots \ldots \ldots \left( vii \right)$
So, the probability that the students take at least one or more subjects would be given as, $P\left( E\cup H\cup S \right)=P\left( E \right)+P\left( H \right)+P\left( S \right)-P\left( E\cap H \right)-P\left( H\cap S \right)-P\left( E\cap S \right)+P\left( E\cap H\cap S \right)$
So, using the equations i, ii, iii, iv, v, vi, and vii, we will get the probability as,
$\begin{align}
  & P\left( E\cup H\cup S \right)=\dfrac{307}{600}+\dfrac{198}{600}+\dfrac{230}{600}-\dfrac{65}{600}-\dfrac{45}{600}-\dfrac{31}{600}+\dfrac{10}{600} \\
 & \Rightarrow P\left( E\cup H\cup S \right)=\dfrac{307+198+230-65-45-31+10}{600} \\
 & \Rightarrow P\left( E\cup H\cup S \right)=\dfrac{604}{600} \\
 & \Rightarrow P\left( E\cup H\cup S \right)=1.0067 \\
\end{align}$
We can see that the probability that the students take one or more subjects is 1.0067, which is greater than 1, but we know that probability can never be greater than 1. Probability is always less than or equal to 1, $P\le 1$.
This was the reason the surveyor was fired as the provided data was incorrect.

Note: While solving the question, the students might not think of using the idea of finding the probability of a student taking 1 or more subjects. Another mistake they could make it in the formula and may subtract the probability of the students who took all the three subjects and write the formula as, $P\left( E\cup H\cup S \right)=P\left( E \right)+P\left( H \right)+P\left( S \right)-P\left( E\cap H \right)-P\left( H\cap S \right)-P\left( E\cap S \right)-P\left( E\cap H\cap S \right)$ and it would lead to a totally incorrect answer.