You are responsible for detecting the source of the error...

You are responsible for detecting the source of the error when a computer system fails. From your analysis you know that the source of error is the disk drive, the computer memory, or the operating system. You know that $50 \%$ of the errors are disk drive errors, $30 \%$ are computer memory errors, and the remainder are operating system errors. From the component performance standards, you know that when a disk drive error occurs, the probability of failure is $0.60$; when a computer memory error occurs, the probability of failure is $0.7$; and when an operating system error occurs, the probability of failure is $0.3$. Given the information from the component performance standards, what is the probability of a disk drive error, given that a failure occurred?

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Key Concepts

Conditional Probability

Conditional probability measures the likelihood that an event will occur given that another event has occurred. In scenarios involving system errors, it is used to quantify the probability of a particular error being the cause when a system failure has been observed.

Bayes' Theorem

Bayes' theorem is a mathematical tool used to update the probability estimate for a hypothesis as additional evidence is acquired. It enables the computation of the probability that a particular error source is responsible for a failure by incorporating prior probabilities and the likelihood of failure given each error source.

Law of Total Probability

The law of total probability is a principle that allows one to calculate the overall probability of an outcome by considering all possible mutually exclusive events that can produce that outcome. In the context of system failures, it is used to determine the overall failure probability by summing the contributions from all potential error sources.

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