### Theory:

Let $$A$$ and $$B$$ are the events of a random  experiment.

Let $$S$$ be the sample space where $$A \subseteq S$$ and $$B \subseteq S$$.

Based on the occurrence of the events $$A$$ and $$B$$, we shall discuss certain algebraic property.

• The event $$(A \cup B)$$:

The events $$(A \cup B)$$ occurs only when at least one of the events $$A$$ or $$B$$ occurs.

This event is depicted using the Venn diagram as follows:

• The event $$(A \cap B)$$:

The events $$(A \cap B)$$ occurs only when both the events $$A$$ and $$B$$ occurs.

This event is depicted using the Venn diagram as follows:

• The event $$\overline A$$:

The event $$\overline A$$ occurs only when the event $$A$$ does not occurs.

This event is depicted using the Venn diagram as follows:

• The event $$\overline B$$:
The event $$\overline B$$ occurs only when the event $$B$$ does not occurs.

This event is depicted using the Venn diagram as follows:

Important!
1. $$P(A \cup \overline A) = S$$
2. $$P(A \cap \overline A) = \phi$$
3. $$\overline {A \cup B} = \overline A \cap \overline B$$ represents the event when neither $$A$$ nor $$B$$ happens.
4. If the events $$A$$ and $$B$$ are mutually exclusive, then $$P(A \cup B) = P(A) + P(B)$$.
5. $$P(\text{Union of events}) = \sum(\text{ Probability of events})$$