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 n independent events occurs, we can use the following steps:

### Step-by-Step Solution:

1. **Understand the Concept of Independent Events**:
- Independent events are those events whose outcomes do not affect each other. The occurrence of one event does not influence the occurrence of another.

2. **Define the Probabilities**:
- Let the probabilities of the n independent events be denoted as \( P_1, P_2, P_3, \ldots, P_n \).

3. **Calculate the Probability of Non-Occurrence**:
- The probability that an event does not occur is given by \( 1 - P_i \) for each event \( i \).
- Therefore, the probabilities of non-occurrence for each event are:
- For event 1: \( 1 - P_1 \)
- For event 2: \( 1 - P_2 \)
- For event 3: \( 1 - P_3 \)
- ...
- For event n: \( 1 - P_n \)

4. **Find the Probability that None of the Events Occur**:
- Since the events are independent, the probability that none of the events occur is the product of their individual probabilities of non-occurrence:
\[ P(\text{none occur}) = (1 - P_1) \times (1 - P_2) \times (1 - P_3) \times \ldots \times (1 - P_n \]

5. **Calculate the Probability that At Least One Event Occurs**:
- The probability that at least one event occurs is the complement of the probability that none of the events occur:
\[ P(\text{at least one occurs}) = 1 - P(\text{none occur}) \]
- Substituting the expression from the previous step:
\[ P(\text{at least one occurs}) = 1 - \left( (1 - P_1) \times (1 - P_2) \times (1 - P_3) \times \ldots \times (1 - P_n) \right) \]

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