The probability of happening an event A in one trial is 0.4. Find the probability that the event A happens at least one in three independent trials.

Explanation

To find the probability that event A happens at least once in three independent trials, we can follow these steps:

### Step 1: Identify the Probability of Event A
The probability of event A happening in one trial is given as:
\[ P(A) = 0.4 \]

### Step 2: Find the Probability of the Complement of Event A
The complement of event A, denoted as \( A' \), is the event that A does not happen. The probability of the complement can be calculated as:
\[ P(A') = 1 - P(A) = 1 - 0.4 = 0.6 \]

### Step 3: Calculate the Probability that Event A Does Not Happen in Three Trials
Since the trials are independent, the probability that event A does not happen in all three trials is:
\[ P(A' \text{ in 3 trials}) = P(A')^3 = (0.6)^3 \]

Calculating \( (0.6)^3 \):
\[ (0.6)^3 = 0.216 \]

### Step 4: Find the Probability that Event A Happens at Least Once in Three Trials
To find the probability that event A happens at least once in three trials, we use the complement rule:
\[ P(\text{A happens at least once}) = 1 - P(A' \text{ in 3 trials}) \]
\[ P(\text{A happens at least once}) = 1 - 0.216 = 0.784 \]

### Final Answer
Thus, the probability that event A happens at least once in three independent trials is:
\[ \boxed{0.784} \]

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