Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit cor

Q: Wait, why do we multiply the probabilities? Shouldn't we add them?

That is a fantastic question! We multiply because we are looking for the probability of a specific *sequence* of events: Plane I misses AND Plane II hits. In probability theory, the 'AND' operator for independent events implies multiplication. If the question asked for the probability that the target is hit by *either* plane, then we would be looking at the union of events, which is where addition comes into play. Here, we are tracing a single, specific path through our decision tree.

Q: Does the probability of Plane II hitting change because Plane I missed?

Not in this problem! The problem states the probabilities of hitting are 0.3 and 0.2 respectively. These are intrinsic to the planes themselves. The fact that Plane I missed is just a *condition* that allows Plane II to take its shot. It doesn't change the skill or accuracy of Plane II. We treat the events as independent in their execution, but sequential in their timing.

Q: What if the question asked for the probability that the target is hit at all?

If the question asked for the total probability of the target being hit, we would need to consider two mutually exclusive scenarios: either Plane I hits, OR Plane I misses and Plane II hits. You would calculate $P(\text{Hit}) = P(I) + (P(\bar{I}) \times P(II))$. It is always helpful to visualize the entire sample space to see how different outcomes relate to each other!

Question

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is

Options:

(A) 0.2

(B) 0.7

(C) 0.06

(D) 0.14

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a command deck, overlooking a target. Two aeroplanes, Plane I and Plane II, are tasked with a mission.

We are given the probabilities of success: $P(I) = 0.3$ and $P(II) = 0.2$.

There is a tactical constraint that changes everything: Plane II will only bomb if Plane I misses. This is the heartbeat of the problem.

### The Anatomy of a Miss

Before we can consider the success of Plane II, we must address the prerequisite. The mission for Plane II is gated by the failure of Plane I.

The probability that Plane I misses, denoted as $P(\bar{I})$, is the complement of it hitting. Mathematically, this is expressed as:

$$P(\bar{I}) = 1 - P(I)$$

Substituting our known value, we get:

$$P(\bar{I}) = 1 - 0.3 = 0.7$$

This $0.7$ represents the seventy percent chance that the first plane fails, thereby clearing the runway for the second plane to engage.

### The Sequential Path

Now, we enter the domain of the second plane. The problem explicitly states that Plane II has a hit probability of $0.2$.

However, this probability is only relevant if the first plane has already missed. We are looking for the probability of a specific intersection of events: the event that Plane I misses AND the event that Plane II hits.

Because these events are sequential and the second is contingent on the first, we apply the multiplication rule. The probability that the target is hit by the second plane is given by:

$$P(	ext{Hit by II}) = P(\bar{I}) 	imes P(II)$$

### Final Calculation

With our logic firmly in place, the calculation becomes a simple, elegant operation. We take our probability of the first miss, $0.7$, and multiply it by the probability of the second plane's success, $0.2$.

$$P(	ext{Hit by II}) = 0.7 	imes 0.2 = 0.14$$

This is the probability that the target is hit specifically by the second plane. The final answer is **0.14**.

It is a testament to how complex, conditional scenarios can be broken down into simple, manageable steps. Remember, in JEE problems, the key is not just the calculation, but the ability to map out the narrative of the events.

Select Answer:

Visualized Solution

The Bombing Scenario

Plane I Success Rate

The Crucial Condition

Probability of Plane I Missing

Evaluating $P (\overset{ˉ}{I})$

Plane II Success Rate

Plane II Miss Rate

The Required Event

Sequential Probability

Multiplication Rule

Substituting Values

Final Answer

The Sigma Insight: Addition and Multiplication Theorems

Analyzing the Setup

The Anatomy of a Miss

The Sequential Path

Final Calculation

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