Problem 74 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The probability that certain electronic component fails when first used is \(0.10 .\) If it does not fail immediately, the probability that it lasts for one year is \(0.99 .\) The probability that a new component will last for one year is (A) \(0.891\) (B) \(0.692\) (C) \(0.92\) (D) none of these

Step-by-Step Solution

Answer

The probability is (A) 0.891.

Step 1: Probability Not Failing Immediately

We start by determining the probability that the component does not fail immediately. This is given as the probability of not failing immediately, which is the complement of the probability of failing immediately. Thus, the probability is \( 1 - 0.10 = 0.90 \).

Step 2: Probability of Lasting One Year Given the Component Does Not Fail Immediately

We know that if the component does not fail immediately, the probability that it lasts for one year is \( 0.99 \).

Step 3: Calculate Overall Probability of Lasting One Year

Now, to find the overall probability that a new component will last for one year, we need to consider both the probability that the component does not fail immediately and the probability that, given it does not fail immediately, it lasts for one year. Hence, the combined probability is: \( 0.90 \times 0.99 = 0.891 \).

Step 4: Compare With Given Options

We compare the calculated probability \( 0.891 \) with the provided options. It matches with option (A).

Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already taken place. This concept is quite useful when we have prior knowledge impacting the outcomes of our events. In the context of this exercise, we are dealing with two specific events:

• Event A: The component does not fail immediately.
• Event B: The component lasts for one year given it did not fail immediately.

Here, the probability of Event B happening, given Event A has already occurred, is known as the conditional probability. This is provided in the problem as 0.99. That means, once we know the component has survived the initial use, there's a 99% chance it will last for a year. Understanding conditional probability helps us analyze sequences of events and helps in making informed decisions in real-world scenarios.

Probability of Failure

Probability of failure represents the chance that a particular component or system does not succeed in performing its intended function. In this example, the electronic component has a certain likelihood of failing right when it's first used. This is quantified as a 0.10 or 10% chance of failure.
The exercise demonstrates how we calculate the probability of failure and use it to understand subsequent outcomes. When calculating probabilities of multiple events, the probability of initial failure plays a critical role. If the component does not fail right away, it continues its operation, leading to other possible outcomes. Grasping this concept aids in predicting the performance and reliability of systems, which is crucial for design and testing phases in engineering and manufacturing.

Probability Calculation

Probability calculation involves determining the likelihood of various outcomes by using mathematical techniques. In our exercise, we first calculate the probability of the component not failing immediately, which is the complement of the failure probability:

• Not failing immediately = 1 - Probability of failure = 1 - 0.10 = 0.90

Next, we incorporate the conditional probability to find the overall probability that the component lasts for a year by multiplying the independent probabilities:

• Probability of lasting one year = (Probability of not failing immediately) \( \times \) (Conditional probability of lasting a year given it doesn't fail immediately) = \( 0.90 \times 0.99 = 0.891 \)

This approach combines multiple probability concepts and demonstrates how these ideas coexist in probability calculations. Probability calculation is fundamental across many industries for risk assessment, quality assurance, and decision-making. By using these calculations, we can estimate outcomes and improve system designs and processes.