### Theory:

Statement:
(i) If $$A$$ and $$B$$ are any two non mutually exclusive events then:

$$P(A \cup B)$$ $$=$$ $$P(A)$$ $$+$$ $$P(B)$$ $$-$$ $$P(A \cap B)$$

(ii) If $$A$$, $$B$$and $$C$$ are any three non mutually exclusive events then:

$$P(A \cup B \cup C)$$ $$=$$ $$P(A)$$ $$+$$ $$P(B)$$ $$+$$ $$P(C)$$ $$-$$ $$P(A \cap B)$$ $$-$$ $$P(B \cap C)$$ $$-$$ $$P(A \cap C)$$ $$+$$ $$P(A \cap B \cap C)$$
Proof of the theorem:
Statement (i):

Let $$A$$ and $$B$$ be any two events of a random experiment.

Let $$S$$ be its sample space.

From the Venn diagram, we observe that the events only $$A$$, only $$B$$ and $$A \cap B$$ are mutually exclusive.

The union of these three events is $$A \cup B$$.

Therefore, $$P(A \cup B)$$ $$=$$ $$P(\text{only }A) + P(A \cap B) + P(\text{only }B)$$.
(i) $$P(A \cap \overline B)$$ $$=$$ $$P(\text{only A})$$ $$=$$ $$P(A) - P(A \cap B)$$

(ii) $$P(\overline A \cap B)$$ $$=$$ $$P(\text{only B})$$ $$=$$ $$P(B) - P(A \cap B)$$
$$\Rightarrow$$ $$P(A \cup B)$$ $$=$$ $$\left[P(A) - P(A \cap B)\right] + P(A \cap B) + \left[P(B) - P(A \cap B)\right]$$

$$\Rightarrow$$ $$P(A \cup B)$$ $$=$$ $$P(A) + P(B) - P(A \cap B)$$

Hence, the proof.

Statement (ii):

Let $$A$$, $$B$$ and $$C$$ be any three events of a random experiment.

Let $$S$$ be its sample space.

Let $$D$$ $$=$$ $$B \cup C$$.

Thus, $$P(A \cup B \cup C)$$ $$=$$ $$P(A \cup D)$$.

$$\Rightarrow$$ $$P(A \cup B \cup C)$$ $$=$$ $$P(A) + P(D) - P(A \cap D)$$ [By statement (i)]

$$=$$ $$P(A) + P(B \cup C) - P\left[A \cap (B \cup C)\right]$$

$$=$$ $$P(A) + \left[P(B) + P(C) - P(B \cap C)\right] - P\left[(A \cap B) \cup (A \cap C)\right]$$

$$=$$ $$P(A) + P(B) + P(C) - P(B \cap C) - \left[P(A \cap B) + P(A \cap C) - P\left((A \cap B) \cap (A \cap C)\right)\right]$$

$$=$$ $$P(A) + P(B) + P(C) - P(B \cap C) - P(A \cap B) - P(A \cap C) + P(A \cap B \cap C)$$

Therefore,
$$P(A \cup B \cup C)$$ $$=$$ $$P(A)$$ $$+$$ $$P(B)$$ $$+$$ $$P(C)$$ $$-$$ $$P(A \cap B)$$ $$-$$ $$P(B \cap C)$$ $$-$$ $$P(A \cap C)$$ $$+$$ $$P(A \cap B \cap C)$$.

Hence, the proof.