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Chapter 2: Problem 167

An optical storage device uses an error recovery procedure that requires an immediate satisfactory readback of any written data. If the readback is not successful after three writing operations, that sector of the disk is eliminated as unacceptable for data storage. On an acceptable portion of the disk, the probability of a satisfactory readback is \(0.98 .\) Assume the readbacks are independent. What is the probability that an acceptable portion of the disk is eliminated as unacceptable for data storage?

Short Answer

Expert verified
The probability is 0.000008.

Step by step solution

01

Understanding the Problem

We need to find the probability that a sector, considered acceptable for data storage, gets eliminated after three unsuccessful readback attempts. We are given that on an acceptable portion, the probability of a satisfactory readback is 0.98.
02

Calculating the Probability of Unsuccessful Readback

The probability of an unsuccessful readback on an acceptable portion is given by the complement of a successful readback. So, \( P(\text{unsuccessful readback}) = 1 - 0.98 = 0.02 \).
03

Applying the Independent Readback Rule

Each readback is independent. Therefore, the probability of having three unsuccessful readbacks in a row is calculated by multiplying the probabilities for each attempt: \( P(3 \, \text{unsuccessful readbacks}) = 0.02 \times 0.02 \times 0.02 \).
04

Final Calculation

Compute the probability from the previous step: \( 0.02 \times 0.02 \times 0.02 = 0.000008 \).
05

Conclusion

This result, 0.000008, is the probability that a sector is eliminated, despite being part of an acceptable portion of the disk.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
The concept of independent events is a cornerstone of probability theory. An event is considered independent if the occurrence or non-occurrence of one event does not influence the probability of another event happening. For example, tossing a coin is independent from rolling a dice: whatever result you get from the coin toss does not affect the dice outcome.

In the context of our exercise, the successful readback attempts are independent of one another. This means that each readback operation鈥檚 success or failure is not affected by the previous try. Thus, the probability of each readback remains constant at every attempt. Understanding independence helps us calculate compound probabilities using the multiplication method more effectively.
Complementary Probability
Complementary probability is a useful concept in probability theory, especially when determining the likelihood of an event not occurring. The idea is simple: the sum of the probabilities of an event and its complement equals 1.

For instance, if the probability of a satisfactory readback is 0.98, the complementary probability, or the chance of a readback being unsuccessful, is calculated as follows:
  • Before readback: Successful = 0.98
  • Therefore, Unsuccessful = 1 - 0.98 = 0.02
This step is crucial in our example exercise because knowing the probability of failure allows us to determine the likelihood of repeated unsuccessful readbacks leading to the sector's elimination.
Multiplication Rule for Probability
The multiplication rule for probability is employed when you need to find the probability of two or more independent events occurring in sequence. This rule states that you can find the compound probability by multiplying the individual probabilities of each event.

For our exercise, we want to find the probability that an acceptable disk sector fails to provide a successful readback across three attempts. Since each readback attempt is independent, we calculate with the multiplication rule:
  • Probability of one failure = 0.02 (as found using complementary probability)
  • Probability of three consecutive failures = 0.02 脳 0.02 脳 0.02 = 0.000008
This demonstrates how the multiplication rule helps determine the overall likelihood of a sequence of independent events resulting in the elimination of a disk sector.

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Most popular questions from this chapter

In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Since acids and bases are usually colorless (as are the water and salt produced in the neutralization reaction), \(\mathrm{pH}\) is measured to monitor the reaction. Suppose that the equivalence point is reached after approximately \(100 \mathrm{~mL}\) of an \(\mathrm{NaOH}\) solution has been added (enough to react with all the acetic acid present) but that replicates are equally likely to indicate from 95 to \(104 \mathrm{~mL},\) measured to the nearest mL. Assume that two technicians each conduct titrations independently. (a) What is the probability that both technicians obtain equivalence at \(100 \mathrm{~mL} ?\) (b) What is the probability that both technicians obtain equivalence between 98 and \(104 \mathrm{~mL}\) (inclusive)? (c) What is the probability that the average volume at equivalence from the technicians is \(100 \mathrm{~mL} ?\)

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