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

S.No. Name Sex Age in years
$1.$ Harish $M$ $30$
$2.$ Rohan $M$ $33$
$3.$ Sheetal  $F$ $46$
$4.$ Alis $F$ $28$
$5.$ Salim $M$ $41$

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?

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Let $E$ be the event in which the spokesperson will be a male and $F$ be the event in which the spokesperson will be over $35$ years of age.

Accordingly, $P ( E )=\frac{3}{5}$ and $P ( F )=\frac{2}{5}$

since there is only one male who is over $35$ years of age,

$P ( E \cap F)=\frac{1}{5}$

We know that $P ( E \cup F)= P ( E )+ P ( F )- P ( E \cap F )$

$\therefore P ( E \cup F )=\frac{3}{5}+\frac{2}{5}-\frac{1}{5}=\frac{4}{5}$

Thus, the probability that the spokesperson will either be a male or over $35$ years of age is $\frac{4}{5}$.

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  • [IIT 2023]

Given that the events $A$ and $B$ are such that $P(A)=\frac{1}{2}, P(A \cup B)=\frac{3}{5}$ and $P(B)=p .$ Find $p$ if they are independent.