Question

Suppose that half of the population of a town is consumers of rice. 100 investigators are appointed to find out its truth. Each investigator interviewed 10 individuals. How many investigators do you expect to report that three or less of the people interviewed are consumers of rice?

Answer 1

Hint: The above problem is a binomial distribution problem in which the probability of success is the number of people consuming rice and it is given that the probability of consuming rice by the individuals from the total population is $\dfrac{1}{2}$ and we have to find the number of investigators given that 3 or fewer people interviewed is consumers of rice. Number of investigators when r people interviewed are consumers of rice are ${}^{10}{{C}_{r}}{{\left( \dfrac{1}{2} \right)}^{r}}{{\left( \dfrac{1}{2} \right)}^{10-r}}.100$. Now, we are going to calculate the number of investigators when 3 or fewer people interviewed are consumers of rice are the addition of the number of investigators when 0, 1, 2 and 3 people interviewed were consumers of rice.

Complete step-by-step solution
It is given that from the given population, the fraction of the population which is consumers of rice is $\dfrac{1}{2}$. And 100 investigators are appointed in which each investigator interview 10 individuals so this is a binomial distribution problem in which the probability that from the r individuals interviewed, all are consumers of rice are:
$P\left( X=r \right)={}^{10}{{C}_{r}}{{\left( \dfrac{1}{2} \right)}^{r}}{{\left( 1-\dfrac{1}{2} \right)}^{10-r}}$
In the above expression, $\dfrac{1}{2}$ is the probability of success which is the ratio of the population which are consumers of the rice from the given population.
Now, the number of investigators when r people interviewed are consumers of rice are:
$E\left( X=r \right)={}^{10}{{C}_{r}}{{\left( \dfrac{1}{2} \right)}^{r}}{{\left( \dfrac{1}{2} \right)}^{10-r}}.100$
The number of investigators who report that 3 or less people interviewed are consumers of rice are:
$E\left( X=0 \right)+E\left( X=1 \right)+E\left( X=2 \right)+E\left( X=3 \right)$
$\begin{align}
  & {}^{10}{{C}_{0}}{{\left( \dfrac{1}{2} \right)}^{0}}{{\left( \dfrac{1}{2} \right)}^{10-0}}.100+{}^{10}{{C}_{1}}{{\left( \dfrac{1}{2} \right)}^{1}}{{\left( \dfrac{1}{2} \right)}^{10-1}}.100+{}^{10}{{C}_{2}}{{\left( \dfrac{1}{2} \right)}^{2}}{{\left( \dfrac{1}{2} \right)}^{10-2}}.100+{}^{10}{{C}_{3}}{{\left( \dfrac{1}{2} \right)}^{3}}{{\left( \dfrac{1}{2} \right)}^{10-3}}.100 \\
 & =1\left( 1 \right){{\left( \dfrac{1}{2} \right)}^{10}}.100+10\left( \dfrac{1}{2} \right){{\left( \dfrac{1}{2} \right)}^{9}}.100+\dfrac{10!}{2!\left( 10-2 \right)!}{{\left( \dfrac{1}{2} \right)}^{2}}{{\left( \dfrac{1}{2} \right)}^{8}}.100+\dfrac{10!}{3!\left( 10-3 \right)!}{{\left( \dfrac{1}{2} \right)}^{3}}{{\left( \dfrac{1}{2} \right)}^{7}}.100 \\
 & =\dfrac{1}{{{2}^{10}}}.100+10\left( \dfrac{1}{{{2}^{10}}} \right).100+\dfrac{10.9.8!}{2.1.8!}\left( \dfrac{1}{{{2}^{10}}} \right).100+\dfrac{10.9.8.7!}{3.2.1.7!}\left( \dfrac{1}{{{2}^{10}}} \right).100 \\
\end{align}$
In the above expression, $8!\And 7!$ will be cancelled out from the numerator and denominator and we get,
$\dfrac{1}{{{2}^{10}}}.100+10\left( \dfrac{1}{{{2}^{10}}} \right).100+\dfrac{10.9}{2.1}\left( \dfrac{1}{{{2}^{10}}} \right).100+\dfrac{10.9.8}{3.2.1}\left( \dfrac{1}{{{2}^{10}}} \right).100$
In the above expression, taking $\dfrac{100}{{{2}^{10}}}$ as common we get,
$\begin{align}
  & \dfrac{100}{{{2}^{10}}}\left( 1+10+45+120 \right) \\
 & =\dfrac{100}{1024}\left( 176 \right) \\
 & =17.19 \\
\end{align}$
Rounding off the above number we get,
17.
Hence, 17 investigators expect to report that three or less of the people interviewed are consumers of rice.

Note: As you can see that the question demands rigorous calculations so make sure you won’t make any calculations mistakes. Along with that while writing the value of ${{2}^{10}}$ generally, one or two “2” will be missed in the multiplication so be careful in writing the power of 2 also.
And another mistake that could happen in the above problem is that you might forget to add the following expression:
${}^{10}{{C}_{0}}{{\left( \dfrac{1}{2} \right)}^{0}}{{\left( \dfrac{1}{2} \right)}^{10-0}}.100$
Because we have to write the number of investigators which you can expect for 3 or fewer number of people interviewed so you might forget to add the expression when no people have been interviewed.