Amit and Nisha appear for an interview for two vacancies in a company. The probability of Amit’s selection is $\dfrac{1}{5}$ and that of Nisha’s selection is $\dfrac{1}{6}$. What is the probability that both of them are selected?

Hint: The probability that both Amit and Nisha are selected in a company is the product of the probability that Amit is selected and the probability that Nisha is selected.

Complete step-by-step solution:
Assume that the probability that Amit is selected in the interview is $P\left( A \right)$ and Nisha is selected in the interview is $P\left( B \right)$. Then the probability that Amit is selected in the interview is given as:
$P\left( A \right) = \dfrac{1}{5}$
The probability that Nisha is selected in the interview is given as:
$P\left( B \right) = \dfrac{1}{6}$
It can be seen that both events are independent events. Therefore, the probability that both of them are selected is given as:
$P\left( {{\text{A and B both selected}}} \right) = P\left( {A \cap B} \right)$
The formula for the independent events is $P\left( {A \cap B} \right) = P\left( A \right) \times P\left( B \right)$.
So, we can rewrite as $P\left( {{\text{A and B both selected}}} \right) = P\left( A \right) \times P\left( B \right)$.
Substitute the value $\dfrac{1}{5}$ for $P\left( A \right)$ and $\dfrac{1}{6}$ for $P\left( B \right)$ in the above formula, to find the probability.
$\begin{array}{c}P\left( {{\text{A and B both selected}}} \right) = \dfrac{1}{5} \times \dfrac{1}{6}\\ = \dfrac{1}{{30}}\end{array}$

Hence, the probability that Amit and Nisha both of them are selected is $\dfrac{1}{{30}}$.

Note: Any two events are said to be an independent event if the occurrence of any one event does not affect the other event. Here, the selection of Amit does not affect the selection of Nisha, therefore, both the events are independent.