Question

From the employees of a company, 5 persons are selected to represents them in the managing committee of the company. Particulars of five persons are as follows:
$$\begin{array}{cccc}\text{S. No.}& \text{Name}& \text{Sex}& \text{Age in years}\\ 1& \text{Harish}& M& 30\\ 2& \text{Rohan}& M& 33\\ 3& \text{Sheetal}& F& 46\\ 4& \text{Alice}& F& 28\\ 5& \text{Salim}& M& 41\end{array}$$
A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over 35 years?

Answer 1

Here total number of persons = 5
One spokesperson is selected out of 5 person in ${}^5{C}_1=5$ ways.
Let A be the event that person is male and B be the event that person is over 35 years.
there are 3 male and one person can be selected in ${}^3{C}_1$ ways.
$\therefore P\left(A\right)=\frac{{}^3{C}_1}{{}^5{C}_1}=\frac{3}{5}$
There are 2 persons who are over 35 years. So one operson can be selected in ${}^2{C}_1$ ways.
$\therefore P\left(B\right)=\frac{{}^2{C}_1}{{}^5{C}_1}=\frac{2}{5}$
There is 1 person who is male and over 35 years.
$\therefore P(A\cap B)=\frac{{}^1{C}_1}{{}^5{C}_1}=\frac{1}{5}$
Now P (either male or over 35 years) =$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)=\frac{3}{5}+\frac{2}{5}-\frac{1}{5}=\frac{3+2-1}{5}=\frac{4}{5}.$