Problem 73 - 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 that a new component lasts for one year is 0.891, so the answer is (A).

Step 1: Understand the Problem

We have two probabilities given: the probability that a component fails immediately is 0.10, and if it doesn't fail immediately (probability of not failing immediately is 0.90), it then has a probability of 0.99 to last for one year. We need to find the probability that a new component lasts for one year.

Step 2: Calculate Immediate Failure

The probability that the component fails immediately is given as 0.10. So, there is a 10% chance that any new component cannot last for a year because it fails when first used.

Step 3: Calculate Probability of Not Failing Initially

The probability of a component not failing immediately, i.e., working initially, is calculated as \(1 - 0.10 = 0.90\). This means there is a 90% chance that the component doesn't fail when first used.

Step 4: Calculate Probability of Lasting One Year Given Initial Success

If the component does not fail immediately, there is a 0.99 chance it will last for one year. Thus, the combined probability that a component doesn't fail initially and lasts for one year is:\(0.90 \times 0.99 = 0.891\).

Step 5: Conclusion

The probability of a new component lasting for one year is the combined probability from when it does not fail immediately. This results in the probability being 0.891.

Key Concepts

Conditional Probability

Conditional probability is a measure of the likelihood that an event will occur, given that another event has already occurred. It provides a way to refine our predictions by incorporating relevant conditions or events.

In the context of the exercise, we have two events:

• The event that an electronic component does not fail immediately (with a probability of 0.90).
• The event that the component lasts for one year, given it did not fail initially, which is 0.99.

Conditional probability allows us to combine these two probabilities to find the overall probability that a component lasts one year, taking into account the condition that it first does not fail. This is expressed as:

• Probability of lasting one year = Probability of not failing initially \( \text{(0.90)} \) times Probability of lasting given no initial failure \( \text{(0.99)} \).

This is a direct application of conditional probability, giving us a combined probability of 0.891 for the component lasting one year.

Failure Rate

Failure rate refers to the frequency with which an engineered system or component fails, expressed in failures per unit of time. In simpler terms, it quantifies how likely a component is to fail when first used, or within a given time frame.

For this exercise, the failure rate is given as 0.10, or 10%, for the electronic component when it is first put to use. This percentage tells us that there's a one in ten chance that a new component will fail immediately.

Understanding the failure rate is crucial for estimating reliability and helps in designing systems more effectively by anticipating possible failures and planning for them. In practice, engineers work to reduce failure rates to ensure that systems are reliable and long-lasting in use.

Probability Calculation

Probability calculation involves determining the likelihood of a specific outcome given one or more possible events. It uses mathematical principles to provide precise answers to questions about uncertainty.

In this problem, our task was to calculate the combined probability that an electronic component does not fail initially and lasts for a year. Here are the steps involved:

• Determine the probability it does not fail initially: \(0.90\).
• Calculate the probability it lasts a year if it does not fail initially: \(0.99\).
• Multiply these probabilities: \(0.90 \times 0.99 = 0.891\).

This computation provides a clear and quantitative answer to the question posed by the exercise. Understanding how to calculate probability helps in making informed decisions based on statistical analysis and expected outcomes. This is particularly useful in many fields, including engineering, finance, and risk management.