Question

In a certain university, the percentage of Hindus, Muslims and Christians among students is $50,25$ and $25$ respectively. If $50\% $ of Hindus, $90\% $ of Muslims and $80\% $ of Christians are smokers, find the probability that a randomly selected smoker student in Muslim.

Answer 1

Hint: In this question we are given the percentage of students of each community and also the percentage of smokers in each community, and by that we can find the total no. of smokers as total no. of students is given and the no. of smokers in each community is also given, so by that we can find the probability that a randomly selected smoker student in Muslim, by using the formula,
$P$ (Finding a Muslim smoker) $=$ $\dfrac{\text{No. of Muslim smokers}}{\text{Total no. of smokers}}$

Complete step-by-step answer:
In this question first of all we will start solving this question by enlisting the given information.
So, it is given in the question that,
 Percentage of Hindus, Muslims and Christians among students is $50,25$ and $25$ respectively in a certain university. $50\% $ of Hindus, $90\% $ of Muslims and $80\% $ of Christians are smokers.
From this information let us assume that total no. of students is $x$
So, we from given information we can say that
No. of Hindus $ = $$50\% $ of $x$
$ \Rightarrow $ No. of Hindus $ = $$\dfrac{{50}}{{100}}\left( x \right)$
$ \Rightarrow $ No. of Hindus $ = $$0.5x$ ……..(i)
No. of Muslims $ = $$25\% $ of $x$
$ \Rightarrow $ No. of Muslims $ = $$\dfrac{{25}}{{100}}\left( x \right)$
$ \Rightarrow $ No. of Muslims $ = $$0.25x$ ……….(ii)
No. of Christians$ = $$25\% $of $x$
$ \Rightarrow $ No. of Christians $ = $$\dfrac{{25}}{{100}}\left( x \right)$
$ \Rightarrow $ No. of Christians $ = $$0.25x$ ………..(iii)
Now, we will calculate the no. of smokers in each community from the given information using (i), (ii) & (iii)
Now, as it is given that $50\% $ of Hindus, $90\% $ of Muslims and $80\% $ of Christians are smokers.
So, No. of Hindu smokers $ = $$50\% $ of No. of Hindus
$ \Rightarrow $ No. of Hindu smokers $ = $$50\% $ of $0.5x$ {putting value of No. of Hindus from equation (i)}
$ \Rightarrow $ No. of Hindu smokers $ = $$\dfrac{{50}}{{100}}\left( {0.5x} \right)$
$ \Rightarrow $ No. of Hindu smokers $ = $$0.5\left( {0.5x} \right) = 0.25x$ …..(iv)
Also, No. of Muslim smokers $ = $$90\% $ of $x$
$ \Rightarrow $ No. of Muslim smokers $ = $$90\% $ of $0.25x$ {putting value of No. of Muslims from equation (ii)}
$ \Rightarrow $ No. of Muslim smokers $ = $$\dfrac{{90}}{{100}}\left( {0.25x} \right)$
$ \Rightarrow $ No. of Muslim smokers $ = $$0.9\left( {0.25x} \right) = 0.225x$ ……....(v)
Similarly, No. of Christian smokers $ = $$80\% $ of $x$
$ \Rightarrow $ No. of Christian smokers $ = $$80\% $ of $0.25x$ {putting value of No. of Christians from equation (ii)}
$ \Rightarrow $ No. of Christian smokers $ = $$\dfrac{{80}}{{100}}\left( {0.25x} \right)$
$ \Rightarrow $ No. of Christian smokers $ = $$0.8\left( {0.25x} \right) = 0.2x$ ……....(vi)
Now as we have derived no. of smokers in each community, so we can find the probability of finding a randomly selected smoker student in Muslim, which could be founded by using the formula
$P$ (Finding a Muslim smoker) $ = $ (No. of Muslim smokers) $/$ (Total no. of smokers)
Here, Total no. of smokers $ = $ No. of Hindu smokers $ + $ No. of Muslim smokers $ + $ No. of Christian smokers
$ \Rightarrow $ As we have proved above that
No. of Hindu smokers $ = $$0.25x$, No. of Muslim smokers $ = $$0.225x$, No. of Christian smokers $ = $$0.2x$
$ \Rightarrow $ By putting the corresponding values we get
Total no. of smokers $ = 0.25x + 0.225x + 0.2x$
$ \Rightarrow $ Total no. of smokers $ = 0.675x$
So, now we will find the probability of Muslim smokers,
$P$ (Finding a Muslim smoker) $ = $ (No. of Muslim smokers) $/$ (Total no. of smokers)
As we have derived above that
No. of Muslim smokers $ = $$0.225x$, Total no. of smokers $ = 0.675x$
Now putting the corresponding values we get
$P$ (Finding a Muslim smoker) $ = \dfrac{{0.225x}}{{0.675x}}$
$ \Rightarrow $ $P$ (Finding a Muslim smoker) $ = \dfrac{1}{3}$

The probability is equal to $\dfrac{1}{3}$.

Note: The alternative ways to do this question is by constructing a Venn diagram from the given information, then firstly find the smokers in each community and then find the probability of Muslim smokers by dividing No. of Muslim smokers by Total no. of smokers.