Question

A special lottery is to be held to select a student who will live in the only deluxe room in a hostel. There are $ 100 $ year-III, $ 150 $ year-II and $ 200 $ year-I students who applied. Each year-III name is placed in the lottery three times; each year-II’s name, 2 times and year-I’s name, $ 1 $ time. What is the probability that a year-III’s name will be chosen?
A. $ \dfrac{1}{8} $
B. $ \dfrac{2}{8} $
C. $ \dfrac{3}{8} $
D. $ \dfrac{1}{2} $

Answer 1

Hint: First we calculate the total number of students by combining all three years students. When we calculate the total number of students we multiply the number of students of year-III by 3 and the number of students of year-II by 2. Then, by using the formula we calculate the desired answer.
Following formula is used-
 $ P\left( A \right)=\dfrac{n\left( E \right)}{n\left( S \right)} $
Where, A is an event,
\[n\left( E \right)=\] Number of favorable outcomes and $ n\left( S \right)= $ number of total possible outcomes

Complete step-by-step answer:
We have given that a special lottery is to be held to select a student who will live in the only deluxe room in a hostel.
We have given that there are $ 100 $ year-III, $ 150 $ year-II and $ 200 $ year-I students who applied.
So, total number of students will be $ 100+150+200=450 $
But we have given that each year-III name is placed in the lottery three times; each year-II’s name, 2 times and year-I’s name, $ 1 $ time.
So, total name in the lottery will be
 $ \begin{align}
  & 100\times 3+150\times 2+200 \\
 & =300+300+200 \\
 & =800 \\
\end{align} $
Now, we have to find the probability that a year-III’s name will be chosen.
We apply the general formula of probability, which is
  $ P\left( A \right)=\dfrac{n\left( E \right)}{n\left( S \right)} $
Now, substituting the values, we get
 $ \begin{align}
  & P\left( A \right)=\dfrac{300}{800} \\
 & P\left( A \right)=\dfrac{3}{8} \\
\end{align} $
So, the probability that a year-III’s name will be chosen is $ \dfrac{3}{8} $
Therefore option C is the correct answer.

Note: The point to be note while solving this question is that we have to multiply by $ 3 $ to the number of students of year-III as given in the question that each year-III name is placed in the lottery three times, multiply by $ 2 $ to the number of students of year-II as given that each year-II’s name is placed 2 times. If we directly calculate the total number of students by simply adding $ 100+150+200 $ , we will get an incorrect answer.