A and B are two events such that \[P\left( A \right) = 0.54\], \[P\left( B \right) = 0.69\] and \[P\left( {A \cap B} \right) = 0.35\]. Find
A). \[P\left( {A \cup B} \right)\]
B). \[P\left( {A' \cap B'} \right)\]
C). \[P\left( {A \cap B'} \right)\]
D). \[P\left( {B \cap A'} \right)\]

Hint: In the given question, we have been given some events. We have been given the probability of occurring in those events. We have to find the probability of the functions on the given event using the former probabilities. We are going to calculate it using the formula of complement of an event, union of an event, and intersection of an event.

Complete step by step solution:
Given, \[P\left( A \right) = 0.54\], \[P\left( B \right) = 0.69\] and \[P\left( {A \cap B} \right) = 0.35\].
(i). Now, we know that \[P\left( {X \cup Y} \right) = P\left( X \right) + P\left( Y \right) - P\left( {X \cap Y} \right)\].
So, putting the values in this value,
\[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\]
or we have,
\[P\left( {A \cup B} \right) = 0.54 + 0.69 - 0.35 = 0.88\]
(ii). We know that \[P{\left( {X \cup Y} \right)^\prime } = P\left( {X' \cap Y'} \right)\]
So, we have,
\[P\left( {A' \cap B'} \right) = 1 - P\left( {A \cup B} \right) = 1 - 0.88 = 0.12\]
(iii). We know that \[P\left( {X \cap Y'} \right) = P\left( X \right) - P\left( {X \cap Y} \right)\]
So, we have,
\[P\left( {A \cap B'} \right) = P\left( A \right) - P\left( {A \cap B} \right) = 0.54 - 0.35 = 0.19\]
(iv).We know that \[P\left( {X \cap Y'} \right) = P\left( X \right) - P\left( {X \cap Y} \right)\]
So, we have,
\[P\left( {A' \cap B} \right) = P\left( B \right) - P\left( {A \cap B} \right) = 0.69 - 0.35 = 0.34\]

Note: In this question, we were given the probability of some events. We had to find the probability of the functions on the given event using the given probabilities. We solved it using various formulae of complement of an event, union of an event, intersection of an event, et cetera. So, it is important that we know the formulae of the probabilities, and where to apply those formulae.