If A and B are two independent events, then prove that the probability that at least one event will happen will be `1-P(A').P(B')`.

Explanation

To prove that the probability that at least one of the independent events A or B occurs is given by the formula \( P(R) = 1 - P(A') \cdot P(B') \), we can follow these steps:

### Step 1: Understand the Events
Let:
- \( A \) be the event that event A occurs.
- \( B \) be the event that event B occurs.
- \( A' \) be the event that event A does not occur (the complement of A).
- \( B' \) be the event that event B does not occur (the complement of B).

### Step 2: Define the Probability of At Least One Event Occurring
The probability that at least one of the events A or B occurs can be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Where \( P(A \cup B) \) is the probability that either A or B occurs.

### Step 3: Use Independence of Events
Since A and B are independent events, the probability of both A and B occurring together is:
\[ P(A \cap B) = P(A) \cdot P(B) \]

### Step 4: Substitute into the Probability Formula
Now we can substitute the expression for \( P(A \cap B) \) into the formula for \( P(A \cup B) \):
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

### Step 5: Express the Complement Events
The probability that event A does not occur is:
\[ P(A') = 1 - P(A) \]
And for event B:
\[ P(B') = 1 - P(B) \]

### Step 6: Find the Probability that Neither Event Occurs
The probability that neither A nor B occurs (both A and B do not happen) is:
\[ P(A' \cap B') = P(A') \cdot P(B') \]
Since A and B are independent, the probability that both do not occur is the product of their individual probabilities:
\[ P(A' \cap B') = (1 - P(A)) \cdot (1 - P(B)) = P(A') \cdot P(B') \]

### Step 7: Final Probability Calculation
Now, the probability that at least one of the events occurs is:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - P(A') \cdot P(B') \]

### Conclusion
Thus, we have shown that:
\[ P(A \cup B) = 1 - P(A') \cdot P(B') \]