You are responsible for quickly fixing your company¡¯s computer system when it crashes. You know that there are three possible sources of errors in the system: the disk drive, the memory and the operating system.
From an analysis of historical data, you find that 50% of the errors are disk errors, 30% are memory errors and the remaining errors are operating system errors.
From component performance standards, you know that when a disk error occurs, the probability that the computer crashes is 60%, when a memory error occurs, the probability that the computer crashes is 70%, and when an operating system error occurs, the probability that the computer crashes is 40%.
The company computer has just crashed! You need to quickly decide which component (disk drive, memory or operating system) is the best component to look at first in your attempt to get the computer back on line.
Which component should you investigate first? Support your decision with a detailed calculation of all of the relevant conditional probabilities.
disk error
To decide which component to investigate first, we need to calculate the conditional probabilities for each component given that the computer has crashed.
Let's denote the events as follows:
D: Disk error
M: Memory error
O: Operating system error
C: The computer crashes
We are given the following probabilities:
P(D) = 0.50
P(M) = 0.30
P(O) = 0.20
The probabilities of the computer crashing given each error are:
P(C|D) = 0.60
P(C|M) = 0.70
P(C|O) = 0.40
We can calculate the probability of each component causing the crash using Bayes' theorem:
P(D|C) = (P(C|D) * P(D)) / (P(C|D) * P(D) + P(C|M) * P(M) + P(C|O) * P(O))
= (0.60 * 0.50) / (0.60 * 0.50 + 0.70 * 0.30 + 0.40 * 0.20)
= 0.30 / (0.30 + 0.21 + 0.08)
= 0.30 / 0.59
= 0.508
P(M|C) = (P(C|M) * P(M)) / (P(C|D) * P(D) + P(C|M) * P(M) + P(C|O) * P(O))
= (0.70 * 0.30) / (0.60 * 0.50 + 0.70 * 0.30 + 0.40 * 0.20)
= 0.21 / (0.30 + 0.21 + 0.08)
= 0.21 / 0.59
= 0.356
P(O|C) = (P(C|O) * P(O)) / (P(C|D) * P(D) + P(C|M) * P(M) + P(C|O) * P(O))
= (0.40 * 0.20) / (0.60 * 0.50 + 0.70 * 0.30 + 0.40 * 0.20)
= 0.08 / (0.30 + 0.21 + 0.08)
= 0.08 / 0.59
= 0.136
Based on these calculations, the highest conditional probability is for a disk error causing the crash with P(D|C) = 0.508. Therefore, the best component to investigate first is the disk drive.