The probability that an event A happens in one trial of an experiment is 0.4. Three independent tria

Q: Wait, why can't we just add the probabilities for each trial? Like $0.4 + 0.4 + 0.4$?

That is a classic trap! Adding probabilities like that assumes the events are mutually exclusive (meaning they cannot happen together), but here, the event $A$ can happen in multiple trials simultaneously. We are dealing with independent trials, so we must use the rules of probability for independent events, not simple addition.

Q: Why do we use $1 - P(\text{none})$? Is there any other way?

You absolutely could calculate $P(X=1) + P(X=2) + P(X=3)$, but that is the long, tedious road. The beauty of the complement rule is that it collapses all those complex cases into one simple subtraction. In JEE, efficiency is just as important as accuracy!

Q: What if the trials were not independent? Would the formula still work?

Great question! If the trials were dependent, the probability of failure in the second trial would depend on what happened in the first. In that case, you could not simply multiply $P(A') \times P(A') \times P(A')$. You would need conditional probabilities. Always check if the problem states 'independent'!

Question

The probability that an event $A$ happens in one trial of an experiment is $0.4$. Three independent trials of the experiment are performed. The probability that the event $A$ happens at least once is

Options:

(A) $0.936$

(B) $0.784$

(C) $0.904$

(D) none of these

Sigma Percentile · Accepted Answer

### The Art of the Complementary Strategy

Welcome, future engineer! Today, we are diving into a classic probability problem that serves as a gateway to mastering the 'at least one' logic—a concept that appears frequently in JEE Advanced papers.

Let us break this down not just as a math problem, but as a strategic challenge.

### The Setup: Visualizing the Experiment

Imagine you are standing before an experiment with three independent slots. In each slot, event $A$ has a probability of occurring, $P(A) = 0.4$.

Naturally, the probability of failure—let us call it $P(A')$—is $1 - 0.4 = 0.6$. We are performing three independent trials. The question asks for the probability that event $A$ happens 'at least once'.

### The Trap: The Long Road

When students first see 'at least once', they often think: 'Okay, that means it happens once, or twice, or three times.' They start calculating $P(X=1) + P(X=2) + P(X=3)$.

While this is mathematically sound, it is a trap of time and complexity. You would have to calculate the binomial probability for each case and sum them up. In the heat of a JEE exam, we want to be faster and smarter.

### The Master Strategy: The Complement

This is where the 'Complementary Event Strategy' shines. Think about the entire sample space of outcomes. Every possible outcome either contains 'at least one success' or it contains 'zero successes'.

There is no middle ground. Therefore, the sum of these two probabilities must be $1$. This gives us the elegant identity:

$$P(	ext{at least one}) = 1 - P(	ext{none})$$

By focusing on the 'none' case, we reduce three complex scenarios into one simple calculation.

### The Execution: Crunching the Numbers

What does 'none' mean? It means event $A$ failed in Trial 1, AND it failed in Trial 2, AND it failed in Trial 3. Because the trials are independent, we can simply multiply the individual probabilities of failure:

$$P(	ext{none}) = P(A') 	imes P(A') 	imes P(A') = 0.6 	imes 0.6 	imes 0.6$$

Calculating this step-by-step: $0.6 	imes 0.6 = 0.36$, and $0.36 	imes 0.6 = 0.216$. Now, we apply our master formula:

$$P(	ext{at least one}) = 1 - 0.216 = 0.784$$

### Final Thoughts

Look at that! We arrived at **$0.784$** with minimal effort. The lesson here is not just about the math; it is about recognizing the structure of the problem.

Whenever you see 'at least one', let your mind immediately pivot to 'one minus none'. It is a powerful tool that will save you precious time and keep you calm under pressure. Keep practicing, keep visualizing, and you will master these concepts in no time!

Animated Solution for Mathematics - Probability: The probability that an event $A$ happens in one trial of an experiment is $0.4$ . Three independent trials of the experiment are performed. The probability that the event $A$ happens at least once is

Select Answer:

Visualized Solution

$P (A)$ and $n$

Probability of Failure $P (A^{'})$

Decoding "At Least Once"

The Complementary Event Strategy

Visualizing "None"

Setting up $P (none)$

Calculating $P (none)$ - Part 1

Calculating $P (none)$ - Part 2

Final Substitution

The Final Subtraction

Conclusion

The Sigma Insight: Binomial Distribution

The Art of the Complementary Strategy

The Setup

The Trap

The Master Strategy

The Execution

Final Thoughts

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