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

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.

Step 1: Identify the Probability of Miss by Plane I

The probability of plane I missing the target is given by 1 minus the probability of hitting the target. Since the probability of hitting the target for plane I is 0.3, the probability of missing is:\[ P(\text{miss by I}) = 1 - 0.3 = 0.7 \]

Step 2: Determine the Situation for Plane II

Since plane II will only bomb the target if plane I misses, the probability that plane II participates is equal to the probability of plane I missing. This probability we calculated in Step 1 as 0.7.

Step 3: Calculate the Probability of Hit by Plane II Given Miss by Plane I

We need to find the probability that plane II hits the target, given that plane I has missed. The probability of plane II hitting the target is 0.2. Therefore, the probability of the target being hit by plane II, given plane I missed, is:\[ P(\text{hit by II}) = P(\text{miss by I}) \times P(\text{hit by II}) = 0.7 \times 0.2 \]

Step 4: Complete the Calculation

Perform the multiplication to find the probability that plane II hits the target after plane I misses:\[ 0.7 \times 0.2 = 0.14 \]

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory. It determines the likelihood of an event occurring, given that another event has already occurred. In our example of aeroplanes bombing a target, plane II only takes action if plane I misses.
Therefore, the occurrence of plane II bombing is dependent on the outcome of plane I missing. This brings into play conditional probability, because we calculate the probability of plane II hitting the target given that plane I has missed it.
The formula for conditional probability is expressed as:\[P(A | B) = \frac{P(A \cap B)}{P(B)}\]where:

• \(P(A|B)\) is the conditional probability of event A occurring given event B has occurred.
• \(P(A \cap B)\) is the probability of both events A and B occurring.
• \(P(B)\) is the probability of event B occurring.

In our scenario, the probability that plane II hits the target \(P(\text{hit by II} | \text{miss by I})\) is calculated by multiplying the probability that I misses by the probability that II hits, providing us with the probability of II hitting under the condition that I has missed.

Independent Events

Independent events are those whose outcomes do not influence each other. In our scenario, though it might seem like the actions of the planes are connected, the probability of the second plane successfully hitting the target is independent of its decision to bomb after plane I misses. Once plane II starts its operation, its probability of hitting the target is an independent event.Consider two independent events, A and B. The probability of both occurring is given by:\[P(A \cap B) = P(A) \times P(B)\]The calculations involving step-by-step probability from the original exercise clarify that, once plane II bombs, its successful hit is independent of the previous action of plane I. However, plane II's participation is conditional upon plane I's miss, highlighting another key aspect in situations involving independent events intersecting with conditional probabilities.

Bayesian Statistics

Bayesian statistics offers a unique approach to probability, building on concepts like conditional probability to update our understanding as we receive more data. It is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. In real-world scenarios, Bayesian methods could adjust our probability estimations as conditions change, or as more data is gathered. Though the example with the aeroplanes doesn't directly use Bayesian formulas, it illustrates an underlying concept where each plane's action informs the others, reflecting Bayesian thinking:

• Each plane's action builds upon the information (i.e., whether the first missed) from the previous actions.
• This updating process of beliefs based on new evidence resonates closely with Bayesian statistics where probabilities are reassessed in a continuous cycle of feedback and adjustment.

Through this iterative process, Bayesian approaches provide powerful tools to deal with uncertainty, especially when cases involve dependent or conditional events.