Problem 43 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 (a) \(0.2\) (b) \(0.7\) (c) \(0.06\) (d) \(0.14\)

Step-by-Step Solution

Answer

The probability that the target is hit by the second plane is \(0.14\). Option (d) is correct.

Step 1: Understanding the Problem

We need to find the probability that the target is hit by the second plane. Let's summarize the conditions: Plane I hits the target with a probability of 0.3. If Plane I misses (probability of 0.7), then Plane II gets a chance to hit it with a probability of 0.2.

Step 2: Calculate Probability of Plane I Missing

We first calculate the probability that Plane I misses the target. If the probability of hitting is 0.3, then the probability of missing is given by: \[ P( ext{I misses}) = 1 - P( ext{I hits}) = 1 - 0.3 = 0.7 \]

Step 3: Conditional Probability of Plane II Hitting

Now, we calculate the probability that Plane II hits given that Plane I has missed. Since Plane II only bombs if Plane I has missed, we use:\[ P( ext{II hits | I misses}) = 0.2 \]

Step 4: Calculate Overall Probability of Plane II Hitting

Combine the probabilities from the previous two steps to find the overall probability that the target is hit by Plane II. This is the product of Plane I missing and Plane II hitting:\[ P( ext{II hits}) = P( ext{I misses}) \times P( ext{II hits | I misses}) = 0.7 \times 0.2 = 0.14 \]

Step 5: Choose the Correct Option

The probability that the target is hit by the second plane is 0.14. Checking the options, we find that option (d) \( 0.14 \) is the correct answer.

Key Concepts

Conditional Probability

Conditional probability is a concept that helps us determine the likelihood of an event occurring, given that another event has already happened. This idea is fundamental when dealing with sequential activities or situations where the outcome of one event influences the conditions of the next.
In the given example, Planes I and II are involved. Plane II only gets the opportunity to bomb if Plane I misses. Thus, to find the probability of Plane II hitting, we calculate the conditional probability, which depends on the condition that Plane I has missed. This is mathematically expressed as:

• \( P(\text{II hits | I misses}) = 0.2 \)

This expression is read as "the probability that Plane II hits given that Plane I misses is 0.2."
Conditional probability allows for a more precise calculation in such scenarios by isolating specific conditions that affect subsequent events.

Independent Events

In probability, independent events refer to situations where the outcome of one event does not influence the outcome of another event. Each event occurs without regard to whether the previous event occurred or not.
However, in the context of this exercise, the bombing events by Planes I and II are not independent. Plane II's action is directly dependent on Plane I's failure to hit. Hence, Plane II's chance to bomb is conditional, which makes them not independent.
For independent events, the probability of both occurring together is the product of their probabilities. But here, since Plane II can only act if Plane I misses, their actions are clearly dependent on one another, necessitating the use of conditional probability.

Probability of Complementary Events

When discussing probability, complementary events consist of all possible outcomes of an event that do not contribute to the occurrence of a desired event. Essentially, the sum of the probabilities of complimentary events equals 1.
In our scenario, the complementary relationship is observed with Plane I's actions. If Plane I hits the target with a probability of 0.3, the complementary event's probability (Plane I missing) is calculated as:

• \( P(\text{I misses}) = 1 - P(\text{I hits}) = 1 - 0.3 = 0.7 \)

Understanding complementary events is crucial as it simplifies the computation of probabilities, especially when dealing with sequential events like in this exercise, and links directly to the concept of conditional probability.