The probability of ‘n’ independent events are `P_(1) , P_(2) , P_(3) ……., P_(n)` . Find an expression for probability that at least one of the events will happen.

Answer

To find the probability that at least one of the independent events occurs, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Problem**: We have 'n' independent events with probabilities \( P_1, P_2, P_3, \ldots, P_n \). We want to find the probability that at least one of these events occurs.

2. **Use the Complement Rule**: The probability that at least one event occurs can be found using the complement rule. The complement of "at least one event occurs" is "none of the events occur".

3. **Calculate the Probability of None Occurring**:
- The probability that the first event does not occur is \( 1 - P_1 \).
- The probability that the second event does not occur is \( 1 - P_2 \).
- Continuing this way, the probability that the nth event does not occur is \( 1 - P_n \).
- Since the events are independent, the probability that none of the events occur is the product of their individual probabilities of not occurring:
\[ P(\text{none occur}) = (1 - P_1) \times (1 - P_2) \times (1 - P_3) \times \ldots \times (1 - P_n \]

4. **Calculate the Probability of At Least One Occurring**:
- Now, using the complement rule:
\[ P(\text{at least one occurs}) = 1 - P(\text{none occur}) \]
- Substituting the expression from the previous step:
\[ P(\text{at least one occurs}) = 1 - \left( (1 - P_1) \times (1 - P_2) \times (1 - P_3) \times \ldots \times (1 - P_n) \right) \]

5. **Final Expression**: Thus, the final expression for the probability that at least one of the events occurs is:
\[ P(\text{at least one occurs}) = 1 - \prod_{i=1}^{n} (1 - P_i) \]