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
$\begin{aligned} & 100\times 3+150\times 2+200 \\\ & =300+300+200 \\\ & =800 \\\ \end{aligned}$
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
$\begin{aligned} & P\left( A \right)=\dfrac{300}{800} \\\ & P\left( A \right)=\dfrac{3}{8} \\\ \end{aligned}$
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.