Problem 63 - Basic Statistics for Business and Economics with Student

Question

A computer-supply retailer purchased a batch of 1,000 CD-R disks and attempted to format them for a particular application. There were 857 perfect CDs, 112 CDs were usable but had bad sectors, and the remainder could not be used at all. a. What is the probability a randomly chosen \(\mathrm{CD}\) is not perfect? b. If the disk is not perfect, what is the probability it cannot be used at all?

Step-by-Step Solution

Answer

a) 0.143, b) 0.217

Step 1: Determine Total Non-perfect CDs

To find the total number of CDs that are not perfect, we need to subtract the number of perfect CDs from the total. There are 1000 CDs in total and 857 perfect ones, so the non-perfect CDs are calculated as:\[1000 - 857 = 143\]

Step 2: Probability of a CD Being Not Perfect

Using the total number of non-perfect CDs calculated previously, the probability of picking a non-perfect CD is the ratio of non-perfect CDs to the total CDs:\[\frac{143}{1000} = 0.143\]

Step 3: Calculate Number of Unusable CDs

The problem states that there are 112 CDs with bad sectors and mentions that the remainder are unusable. Therefore, the number of unusable CDs is:\[143 - 112 = 31\]

Step 4: Calculate Probability a CD is Unusable Given it's Not Perfect

Given that a CD is not perfect, the probability that it cannot be used is the ratio of unusable CDs to the total non-perfect CDs:\[\frac{31}{143} \approx 0.217\]

Key Concepts

CD-R disks

When we talk about CD-R disks, we're essentially discussing a type of optical disc that can store digital data. These disks are popular because they are relatively inexpensive and can hold large amounts of information. CD-R stands for 'Compact Disc-Recordable', which means once you record data on them, it usually can't be changed. For students and businesses alike, these disks often serve as a handy solution for data storage.

In a batch of CD-Rs, there can be variations in quality, which might affect their usability. Some CD-Rs might be flawless, while others may contain minor imperfections. Understanding the nature of these imperfections is crucial for anyone dealing with large quantities of CD-Rs, like retailers or consumers who use CDs for backups and data transfer.

Non-perfect CDs

In any production batch, 'non-perfect CDs' refer to those disks that do not meet the standard criteria of being flawless. This might be due to imperfections or defects that affect data recording or retrieval. In the provided scenario, out of 1,000 CD-R disks, 143 CDs fall under the category of non-perfect.

Such CDs can still be usable, but their imperfections, like bad sectors, might pose challenges. Bad sectors are areas on the disk where data can’t be read or written reliably. Consequently, while these disks are considered non-perfect, they can still function to some degree. It’s essential to identify and manage these poorly performing disks to minimize disruptions in data operations.

Unusable CDs

Unusable CDs are those that can't be utilized at all, even if they are part of the non-perfect category. In our situation, the batch contained 31 completely unusable CDs. These are the units that, due to significant defects or damages, cannot hold data reliably or at all.

Understanding and calculating unusable CDs helps businesses manage their inventory better and adjust their purchasing and quality control processes. In this exercise, the calculation reveals that these CDs form a smaller subset of non-perfect CDs, emphasizing the importance of quality checks before usage.

Calculating probabilities

Calculating probabilities involves determining the likelihood of certain outcomes occurring, which is an essential aspect of statistics. For the CD-R disks in this scenario, there are two main probabilities to compute:

• The probability that a CD is not perfect, calculated as the number of non-perfect CDs divided by the total CDs, which equals 0.143 or 14.3%.
• The probability that a CD is unusable given it's non-perfect, calculated by dividing the number of unusable CDs by the total number of non-perfect CDs, resulting in approximately 0.217, or 21.7%.

These probabilities are crucial for decision-making, such as determining whether to order a different batch or address quality issues with suppliers.

In real-life applications, understanding these probabilities aids in evaluating risks and making informed business choices. It involves logical reasoning and quantitative skills, contributing to effective data handling and operational efficiency.