<p>The probability of happening an event <i>A</i> in one trial is 0.4. Find the probability that the event <i>A</i> happens at least once in three independent trials.</p>
Step-by-Step Solution
Key Concept: Use the complement principle: P(at least once) = 1 - P(never happens). It's faster and avoids calculating multiple cases.
<p><strong>Step 1:</strong> Identify given information</p><p>P(A happens in one trial) = p = 0.4</p><p>P(A doesn't happen in one trial) = q = 1 - 0.4 = 0.6</p><p>Number of trials = n = 3</p><p><strong>Step 2:</strong> Apply complement principle for 'at least once'</p><p>P(A happens at least once) = 1 - P(A never happens)</p><p><strong>Step 3:</strong> Calculate P(A never happens in 3 trials)</p><p>Since trials are independent:</p><p>P(A never happens) = P(A' in trial 1) × P(A' in trial 2) × P(A' in trial 3)</p><p>P(A never happens) = (0.6) × (0.6) × (0.6) = (0.6)³ = 0.216</p><p><strong>Step 4:</strong> Find final probability</p><p>P(at least once) = 1 - 0.216 = <strong>0.784</strong></p>
Correct Answer: 0.784