Question

Out of $13$ applicants for a job, there are $5$ women and $8$ men. It is desired to select $2$ persons for the job. The probability that at least one of the selected persons will be a woman is:

• A. $\frac{25}{39}$
• B. $\frac{14}{39}$
• C. $\frac{5}{13}$
• D. $\frac{10}{13}$

Answer 1

The correct option is A

$\frac{25}{39}$

Explanation for correct option

Calculating the probability of selecting $2$ people for the job:

Given, the total number of applicants =$13$

Number of women = $5$

Number of men = $8$

Probability of selecting any two random people for the job the job$={}^{13}\mathrm{C}_2$

Probability of selecting two people for the job when one is a woman and another is a man:

$={}^5{C}_1{\times }^8{C}_1$

Probability of selecting two people for the job when both are women:

$={}^5{C}_2$

So, Favourable cases $={}^5\mathrm{C}_1\times {}^8\mathrm{C}_1+{}^5\mathrm{C}_2$

Therefore, the required probability of selecting two people that alteast one of them is a woman:

$$\begin{gathered} =\frac{{}^5\mathrm{C}_1\times {}^8\mathrm{C}_1+{}^5\mathrm{C}_2}{{}^{13}\mathrm{C}_2} \\ =\frac{40+10}{78} \\ =\frac{25}{39} \end{gathered}$$

Hence, the correct answer is option $\left(\mathrm{A}\right)$.