Question

From a city population, the probability of selecting a male or smoker is $\frac{7}{10}$, a male smoker is $\frac{2}{5}$ and a male, if a smoker is already selected, is $\frac{2}{3}$. Then, Probability of selecting a smoker, if a male is first selected

• A. $\frac{3}{2}$
• B. $\frac{1}{5}$
• C. $\frac{4}{5}$
• D. $\frac{8}{5}$

Answer 1

The correct option is C $\frac{4}{5}$

Suppose $A:$ a male is selected
$B:$ a smoker is selected
Given:
$P(A\cup B)=\frac{7}{10},P(A\cap B)=\frac{2}{5}$ and $P\left(\frac{A}{B}\right)=\frac{2}{3}$
The probability of selecting a smoker.
$P\left(B\right)=\frac{P(A\cap B)}{P\left(\frac{A}{B}\right)}$
$=\frac{2\times 3}{5\times 2}=\frac{3}{5}$
The probability of selecting a non-smoker
So, $P\left(B\right)=1-P\left(B\right)$ ... [using conditional probability]
$=1-\frac{3}{5}=\frac{2}{5}$
The probability of selecting a male
$P\left(A\right)=P(A\cup B)+P(A\cap B)-P\left(B\right)$
$=\frac{7}{10}+\frac{2}{5}-\frac{3}{5}$
$=\frac{7+4-6}{10}=\frac{1}{2}$
Probability of selecting a smoker, if a male is first selected, is given by
$P\left(\frac{B}{A}\right)=\frac{P(A\cap B)}{P\left(A\right)}$ ..... [Using conditional probability]
$=\frac{2}{5}\times \frac{2}{1}=\frac{4}{5}$

Hence, only option C is correct.