The probability that an event A happens in one trial of an experiment, is 0.4 There independent trials of the experiments are performed. The probability that the event A happens atleast once, is

Options

A. 0.936
B. 0.784
C. 0.904
D. None of these

Answer

To solve the problem, we need to find the probability that event A happens at least once in three independent trials, given that the probability of A happening in one trial is 0.4.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Probability of event A happening in one trial, \( P(A) = 0.4 \)
- Probability of event A not happening in one trial, \( P(A') = 1 - P(A) = 1 - 0.4 = 0.6 \)
- Number of trials, \( N = 3 \)

2. **Understanding the Probability of "At Least Once"**:
- The probability that event A happens at least once in three trials can be calculated using the complement rule.
- The complement of "at least once" is "not at all", which means event A does not happen in any of the trials.

3. **Calculate the Probability of Event A Not Happening in All Trials**:
- The probability that event A does not happen in all three trials is given by:
\[ P(A' \text{ in all 3 trials}) = P(A')^N = (0.6)^3 \]

4. **Calculate \( (0.6)^3 \)**:
- Calculate \( (0.6)^3 \):
\[ (0.6)^3 = 0.6 \times 0.6 \times 0.6 = 0.216 \]

5. **Use the Complement Rule**:
- Now, we can find the probability that event A happens at least once:
\[ P(A \text{ happens at least once}) = 1 - P(A' \text{ in all 3 trials}) = 1 - 0.216 \]

6. **Final Calculation**:
- Calculate \( 1 - 0.216 \):
\[ 1 - 0.216 = 0.784 \]

### Conclusion:
The probability that event A happens at least once in three trials is \( \boxed{0.784} \).