For any three events A,B andC,`P(AuuBuuC)` =……………….

Answer

To find the probability of the union of three events A, B, and C, we can use the principle of inclusion-exclusion. The formula for the probability of the union of three events is given by:

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

### Step-by-step Solution:

1. **Identify the individual probabilities**: Start by identifying the probabilities of each event A, B, and C. This means you need to know \( P(A) \), \( P(B) \), and \( P(C) \).

2. **Calculate the pairwise intersections**: Next, find the probabilities of the intersections of each pair of events:
- \( P(A \cap B) \)
- \( P(B \cap C) \)
- \( P(A \cap C) \)

3. **Calculate the intersection of all three events**: Find the probability of the intersection of all three events:
- \( P(A \cap B \cap C) \)

4. **Apply the inclusion-exclusion principle**: Substitute all the identified probabilities into the inclusion-exclusion formula:
\[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C) \]

5. **Simplify the expression**: Perform the arithmetic to simplify the expression if necessary.

### Final Formula:
Thus, the final formula for the probability of the union of three events A, B, and C is:
\[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C) \]