If P(A)=0.3, P(B)=0.2 and P(C)=0.1 and A,B,C are independent events, find the probability of occurrence of at least one of the three events A,B and C.

Answer

To find the probability of occurrence of at least one of the three independent events A, B, and C, we can use the complement rule. The complement of at least one of the events occurring is that none of the events occur.

Let's denote the probabilities as follows:
- P(A) = 0.3
- P(B) = 0.2
- P(C) = 0.1

### Step-by-Step Solution:

1. **Calculate the probability of none of the events occurring**:
Since A, B, and C are independent events, the probability of none of them occurring is given by:
\[ P(\text{none of A, B, C}) = P(A') \times P(B') \times P(C') \]
where \( P(A') \), \( P(B') \), and \( P(C') \) are the probabilities of A, B, and C not occurring, respectively.

We can calculate these as:
\[ P(A') = 1 - P(A) = 1 - 0.3 = 0.7 \]
\[ P(B') = 1 - P(B) = 1 - 0.2 = 0.8 \]
\[ P(C') = 1 - P(C) = 1 - 0.1 = 0.9 \]

2. **Multiply the probabilities of the complements**:
Now we can calculate:
\[ P(\text{none of A, B, C}) = P(A') \times P(B') \times P(C') = 0.7 \times 0.8 \times 0.9 \]

Performing the multiplication:
\[ 0.7 \times 0.8 = 0.56 \]
\[ 0.56 \times 0.9 = 0.504 \]

Thus, \( P(\text{none of A, B, C}) = 0.504 \).

3. **Calculate the probability of at least one event occurring**:
Finally, we can find the probability of at least one of the events A, B, or C occurring:
\[ P(\text{at least one of A, B, C}) = 1 - P(\text{none of A, B, C}) = 1 - 0.504 = 0.496 \]

### Final Answer:
The probability of occurrence of at least one of the three events A, B, and C is **0.496**.