\[A,B\] and \[C\] are any three events. If \[P\left( S \right)\] denotes the probability of \[S\] happening. Then what is the value of \[P\left( {A \cap \left( {BUC} \right)} \right)\]?
A. \[P\left( A \right) + P\left( B \right) + P\left( C \right) - P\left( {A \cap B} \right) - P\left( {A \cap C} \right)\]
B. \[P\left( A \right) + P\left( B \right) + P\left( C \right) - P\left( B \right)P\left( C \right)\]
C. \[P\left( {A \cap B} \right) + P\left( {A \cap C} \right) - P\left( {A \cap B \cap C} \right)\]
D. None of these

Hint: First we will apply the formula \[\left( {A \cap \left( {B \cup C} \right)} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right)\]in \[P\left( {A \cap \left( {B \cup C} \right)} \right)\]. Then we will apply \[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\] in \[P\left( {\left( {A \cap B} \right) \cup \left( {A \cap C} \right)} \right)\]. After that we will apply \[\left( {A \cap B} \right) \cap \left( {A \cap C} \right) = \left( {A \cap B \cap C} \right)\] to get desire result.

Formula used:
1. \[\left( {A \cap \left( {BUC} \right)} \right) = \left( {A \cap B} \right)U\left( {A \cap C} \right)\]
2. \[\left( {A \cap B} \right) \cap \left( {A \cap C} \right) = \left( {A \cap B \cap C} \right)\]
3. \[P\left( {AUB} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\]

Complete step by step solution:
Given: \[A,B\] and \[C\] are any three events.
Let’s simplify the probability \[P\left( {A \cap \left( {BUC} \right)} \right)\].
Now apply the formula \[\left( {A \cap \left( {BUC} \right)} \right) = \left( {A \cap B} \right)U\left( {A \cap C} \right)\].
\[P\left( {A \cap \left( {BUC} \right)} \right) = P\left( {\left( {A \cap B} \right)U\left( {A \cap C} \right)} \right)\]

Expand the above equation using the formula \[P\left( {AUB} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\].
\[P\left( {A \cap \left( {BUC} \right)} \right) = P\left( {A \cap B} \right) + P\left( {A \cap C} \right) - P\left( {\left( {A \cap B} \right) \cap \left( {A \cap C} \right)} \right)\]

Now apply the formula \[\left( {A \cap B} \right) \cap \left( {A \cap C} \right) = \left( {A \cap B \cap C} \right)\].
\[P\left( {A \cap \left( {BUC} \right)} \right) = P\left( {A \cap B} \right) + P\left( {A \cap C} \right) - P\left( {A \cap B \cap C} \right)\]
Hence the correct option is C.

Note: Sometimes students apply the formula \[P\left( {\left( {A \cap B} \right) \cup \left( {A \cap C} \right)} \right) = P\left( {A \cap B} \right) + P\left( {A \cap C} \right) - P\left( {\left( {A \cap B} \right) \cup \left( {A \cap C} \right)} \right)\]. But the correct formula is \[P\left( {\left( {A \cap B} \right) \cup \left( {A \cap C} \right)} \right) = P\left( {A \cap B} \right) + P\left( {A \cap C} \right) - P\left( {\left( {A \cap B} \right) \cap \left( {A \cap C} \right)} \right)\].