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In the realm of probability theory, understanding the concept of independent events is crucial. Independent events are those whose outcomes do not affect one another. This means the occurrence of one event neither increases nor decreases the probability of the other event occurring. In simpler terms, each event is self-contained and has its own inherent probability, unaffected by other events.

In the context of our exercise, each readback attempt is considered an event. We assume these are independent because the success or failure of one readback does not influence the success or failure of subsequent readbacks. This means that regardless of what happened in previous write attempts, the probability of a successful readback remains constant at 0.98 for each attempt. Understanding this independence is key to accurately calculating the overall probability of failing a series of readbacks.

Complementary probability helps us tackle problems by looking at the flip side of an event. In probability, the complement of an event is simply what happens when the event does not occur. For example, if the probability of an event happening is given, its complement–the probability of the event not happening–can be derived easily.

In our exercise, the probability of a successful readback is 0.98. To find the probability of an unsuccessful readback, we compute the complement:

• Successful readback probability: 0.98
• Unsuccessful readback probability: 1 - 0.98 = 0.02

This complementary probability is vital for determining the likelihood of an event that is composed of multiple independent trials, especially when we want to find the probability of a series of unsuccessful outcomes.

Calculating probabilities involves understanding how to use basic operations in probability theory to find the likelihood of events occurring. In our example, the calculation of probabilities helps us determine the chance that an acceptable portion of the disk becomes unacceptable due to failed readbacks.

After determining the complementary probability of an unsuccessful readback at 0.02, we move to find the probability of three unsuccessful readbacks in a row. This kind of problem is solved by multiplying the probability of each independent unsuccessful event:

• First unsuccessful readback: 0.02
• Second unsuccessful readback: 0.02
• Third unsuccessful readback: 0.02

The combined probability for all three readbacks to fail is \[ (0.02) imes (0.02) imes (0.02) = (0.02)^3 \]Computing \[ (0.02)^3 \],we get the final result: 0.000008.

This final probability represents how likely it is for an initially acceptable disk sector to be deemed unacceptable after three consecutive failures. Through these calculations, we gain insight into how probabilities are composed and determined, particularly when dealing with independent events and their complements.