Question

From the employees of a company, 5 persons are selected to represent them in the managing committee of the company. Particulars of the five persons are as follows:

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

Hint: We have been given a dataset and we have to find the probability of an event, therefore we can apply the definition of probability and the probability of the union of the events to obtain the required answer.

Complete step-by-step answer:

We know that the law of probability is given by
$\text{Probability of occurence of an event A=}\dfrac{\text{No}\text{. of events favourable to A}}{\text{Total number of events}}..............(1.1)$
Where the favorable events of a correspond to those events for which the conditions for an event belonging to A are satisfied.
We find that the total number of employees is 5. Thus the total number of possible events while selecting an employee is 5.
Let A be the event corresponding to an employee being a male and B be the event corresponding to an employee having age over 35 years.
The number of male employees is 3 (Harish, Rohan and Saleem). Thus, from equation (1.1)
$\text{Probability of selecting a male employee=P(A)=}\dfrac{3}{5}$
The number of employees having age over 35 is 2 (Sheetal and Saleem). Therefore, from equation (1.1)
$\text{Probability of selecting an employee having age over 35=P(B)=}\dfrac{2}{5}$

$P(A\cap B)$is the probability that the employee is a male and is also aged above 35 years which is satisfied by 1 person, Saleem.
Therefore, from equation (1.1)
$P(A\cap B)=\dfrac{1}{5}$
The probability that the spokesperson will be either male or over 35 years will be given by the probability of the union of the events A and B.
By probability theory, the probability of the union of two events A and B is given by
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Therefore,
$\begin{aligned} & \text{Probability of selecting an employee being a male or having age over 35 years} \\\ & \text{=P(A)+P(B)-P(A}\cap \text{B)=}\dfrac{2}{5}+\dfrac{3}{5}-\dfrac{1}{5}=\dfrac{4}{5} \\\ \end{aligned}$
Thus, the required answer is equal to $\dfrac{4}{5}$.

Note: We should be careful to subtract the probability of obtaining the intersection of the events from the total probability as otherwise the probability of the event corresponding to the intersection of the events would be added twice and thus give the incorrect result.