Question

In a certain university , the percentage of a Hindus, Muslim and Christians among student are 50, and 25 and 25 respectively. if 50% of Hindus, 90% of Muslim and 80% of Christians are smokers, Find the probability that a randomly selected smoker student in Muslim.

Answer 1

Consider the problem

Let $E_1,E_2,E_3$ be the event of choosing a hindu, Muslim and christian respectively.

And Let $E$ be the event that the selected student smoke.

Then,

$\begin{array}{c}p\left(E_1\right)=\frac{50}{100}=\frac{1}{2}\\ p\left(E_2\right)=\frac{25}{100}=\frac{1}{4}\\ p\left(E_3\right)=\frac{25}{100}=\frac{1}{4}\end{array}$

And

$\begin{array}{c}p\left(\frac{E}{E_1}\right)=\frac{50}{100}\\ p\left(\frac{E}{E_2}\right)=\frac{90}{100}\\ p\left(\frac{E}{E_3}\right)=\frac{80}{100}\end{array}$

$\begin{array}{c}p\left(\frac{E_2}{E}\right)=\frac{p\left(E_2\right)\times p\left(\frac{E}{E_2}\right)}{p\left(E_1\right)p\left(\frac{E}{E_1}\right)+p\left(E_2\right)p\left(\frac{E}{E_2}\right)+p\left(E_3\right)p\left(\frac{E}{E_3}\right)}\\ =\frac{\frac{25}{100}\times \frac{90}{100}}{\frac{50}{100}\times \frac{50}{100}+\frac{25}{100}\times \frac{90}{100}+\frac{25}{100}\times \frac{80}{100}}\\ =\frac{25\times 90}{50\times 50+25\times 90+25\times 80}\\ =\frac{90}{2\times 50+90+80}\\ =\frac{45}{50+45+40}\\ =\frac{45}{135}\\ =\frac{1}{3}\end{array}$