Question

The failure rate for a guided missile control system is 1 in \(1000 .\) Suppose that a duplicate, but completely independent, control system is installed in each missile so that, if the first fails, the second can take over. The reliability of a missile is the probability that it does not fail. What is the reliability of the modified missile?

Short answer

Answer: The reliability of the modified missile with two independent control systems is $$\frac{999999}{1000000}$$.

Step-by-step solution

Calculate the failure and reliability of a single control system

The failure rate of a single control system is given as 1 in 1000, meaning it has a probability of failing $$\frac{1}{1000}$$. To find its reliability (the probability of success), we can subtract this probability from 1. Reliability of a single control system = $$1 - \frac{1}{1000} = \frac{999}{1000}$$

Calculate the probability of both control systems failing simultaneously

Since the two control systems are independent, we can find the probability of both failing by multiplying the failure probability of each control system. Probability of both control systems failing = $$\frac{1}{1000} \times \frac{1}{1000} = \frac{1}{1000000}$$

Calculate the reliability of the modified missile

To find the reliability of the modified missile, we need to find the probability that at least one control system works correctly. This is equal to 1 minus the probability that both control systems fail. Reliability of the modified missile = $$1 - \frac{1}{1000000} = \frac{999999}{1000000}$$ So, the reliability of the modified missile with two independent control systems is $$\frac{999999}{1000000}$$.

Key concepts

Independent Events

In a reliability assessment of systems, understanding independent events is crucial. Two events are considered independent if the occurrence of one does not affect the probability of the other occurring. In the context of the missile control system, each control setup is an independent event. This means that the failure of one system does not alter the likelihood of the other one failing.

When dealing with probabilities, if you're dealing with independent events like these control systems, you calculate the probability of both events happening (e.g., both systems failing) by multiplying the probabilities of each event. This multiplication rule makes it much simpler to evaluate the overall system reliability by examining each component separately. In this missile control scenario, we're considering the chance that both systems fail simultaneously which forms the basis for understanding how the system's overall reliability improves when redundancy is added.

Probability of Failure

Probability of failure is a fundamental concept when analyzing any system's reliability. It quantifies the likelihood that a system will not perform its intended function. In probabilistic terms, it is expressed as a value between 0 and 1. In our missile control example, the failure rate of one system is given as \( \frac{1}{1000} \). This represents a low probability, indicating high reliability of the individual control unit.

To understand the risk of system failure more deeply, we look at the combined failure probability of multiple independent systems. By calculating the simultaneous failure probability of these systems, \( \frac{1}{1000} \times \frac{1}{1000} = \frac{1}{1000000} \), we can see that adding more components substantially reduces failure probability. This practice forms the basis for enhancing system reliability by designing redundant systems, where additional components are added to back up the primary ones.

Redundancy in Systems

Redundancy in systems is the practice of incorporating extra components that are not strictly necessary for functionality but serve to increase reliability. In the missile system scenario, redundancy is achieved by installing a duplicate control system. This backup kicks in if the first system fails.

Redundancy boosts a system's reliability by providing alternative pathways for the system to succeed. By reducing the likelihood of complete failure, reliability calculations often improve markedly. With two independent systems working in conjunction, the probability of both failing drops significantly, making the overall system much more robust. As seen in the missile example, the redundancy dramatically improves reliability from \( \frac{999}{1000} \) for a single system to \( \frac{999999}{1000000} \) for the modified, redundant system. This practice of using redundancy is crucial in critical systems where failure can have significant consequences.