Problem 161 - A Complete Resource Book in Mathematics for JEE Main 2017

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 \([2007]\)(A) \(0.06\) (B) \(0.14\) (C) \(0.2\) (D) \(0.7\)

Step-by-Step Solution

Answer

The probability that the target is hit by the second plane is 0.14; option (B) is correct.

Step 1: Determine the Probability of Plane I Missing

The probability that Plane I misses the target is the complement of its probability to hit the target. Since Plane I has a probability of 0.3 to hit the target, the probability of it missing is expressed as follows:\[ P(M_1) = 1 - 0.3 = 0.7 \]

Step 2: Calculate the Combined Probability for Plane II Hitting After I Misses

The second plane will bomb the target only if the first plane misses. The probability of the second plane successfully hitting the target given that Plane I has missed can be calculated by multiplying the probability of the first plane missing by the probability of the second plane hitting the target.\[ P(H_2 | M_1) = P(M_1) \times P(H_2) = 0.7 \times 0.2 = 0.14 \]

Step 3: Conclusion: Select the Correct Option

Based on the calculation in Step 2, the probability that the target is hit by the second plane is 0.14. Therefore, the correct option is: (B) 0.14.

Key Concepts

Conditional Probability

Conditional probability is a concept that allows us to determine the likelihood of an event occurring, given that another event has already happened. In the context of the exercise, conditional probability is crucial because the second plane only takes action based on whether the first plane hits or misses the target. This dependency directly shapes the probability calculations.
To calculate the conditional probability of the target being hit by Plane II, given that Plane I has missed, you need to understand how these events are linked. The formula used here is \( P(H_2 | M_1) = P(M_1) \times P(H_2) \), which combines the chance of Plane I missing, \( P(M_1) \), with Plane II hitting the target, \( P(H_2) \).
This approach highlights how past events (Plane I missing) influence the subsequent events (Plane II potentially hitting). Such calculations are essential in scenarios with dependent events, such as reliability assessments and strategic planning.

Complementary Events

To fully grasp the exercise, understanding complementary events is vital. Complementary events are pairs of events where the occurrence of one event means the other cannot happen; they are mutually exclusive. In probability terms, if an event's probability is \( P(A) \), then the probability of its complement is \( 1 - P(A) \).
In this exercise:

• Event A: Plane I hits the target with \( P(H_1) = 0.3 \)
• Complement of A: Plane I misses the target, calculated as \( P(M_1) = 1 - 0.3 = 0.7 \)

By identifying the complement, we determine Plane II's opportunity to take action. This approach simplifies problems involving sequential events, by focusing on what happens if the first event does not occur.

Probability of Independent Events

Understanding independent events is essential to solving problems involving multiple scenarios. Independent events are those where the outcome of one does not affect the other. However, in this exercise, it's critical to recognize when events are not independent.
While Plane I and Plane II's actions are sequentially dependent (because Plane II acts only if Plane I misses), the probability of Plane I hitting or missing the target is independent of whether Plane II hits, given that Plane I missed. This distinction shapes how probabilities are calculated.
Assessing the probability of Plane II hitting the target involves recognizing its standalone hit probability \( P(H_2) = 0.2 \) and combining it with the condition that Plane I must have missed first. Here, recognizing when events are independent or dependent is key to ensuring accurate probability calculations across various scenarios.