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Chapter 23: Problem 1

A company manufactures precision bearings. On an inspection 3 batches out of 500 were rejected. Calculate the probability that a batch is rejected.

Short Answer

Expert verified
The probability that a batch is rejected is 0.006 or 0.6\%.

Step by step solution

01

Identify Given Information

The problem provides that 3 out of 500 batches were rejected. This means 3 batches do not meet the quality standard out of a total of 500 batches.
02

Understand Probability Formula

The probability of an event occurring is determined by dividing the number of favorable outcomes by the total number of possible outcomes. Here, the 'favorable' outcome is rejecting a batch.
03

Apply Probability Formula

To find the probability of a batch being rejected, use the formula: \( P(\text{rejected}) = \frac{\text{Number of rejected batches}}{\text{Total number of batches}} \). Substitute the known values: \( P(\text{rejected}) = \frac{3}{500} \).
04

Simplify the Fraction

Divide 3 by 500 to get the probability in decimal form: \( \frac{3}{500} = 0.006 \).
05

Express the Probability as a Percentage

Convert the decimal probability into a percentage by multiplying by 100: \( 0.006 \times 100 = 0.6\% \).

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Key Concepts

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

Probability Formula
In the realm of probability and statistics, the Probability Formula is a foundational concept. It allows us to measure the likelihood of an event occurring. The formula is straightforward:
  • Probability (P) = Number of Favorable Outcomes / Total Number of Possible Outcomes
For instance, in the manufacturing exercise we are considering, the favorable outcome is a rejected batch. To calculate the probability, we divide the number of rejected batches by the total number of batches.
This method gives us a clear, quantifiable measure of how likely a rejected batch is within a manufacturing process.
Decimal to Percentage Conversion
Once you have calculated a probability in decimal form, it's often useful to convert it into a percentage. This step makes it easier to understand and communicate the probability.
To perform this conversion, multiply the decimal by 100, and append a '%' symbol. Deriving percentages from decimals offers a more intuitive grasp of the data, particularly in areas like quality control.
Let's look at our example: converting the probability of a rejected batch, 0.006, into a percentage involves simply performing this operation:
  • The calculation goes like this: 0.006 脳 100 = 0.6%
A 0.6% defect rate is much more tangible and easy to convey than a 0.006 probability when discussing quality metrics.
Statistical Analysis
Statistical Analysis plays a pivotal role in understanding data patterns, especially in industrial settings. It involves collecting, analyzing, and interpreting large amounts of data to make informed decisions.
In our exercise, evaluating the rejection rates of batches can help identify trends or problems in the production line.
  • Helps pinpoint where and why defects occur
  • Allows for the implementation of strategies to reduce batch rejections
By continuously analyzing statistical data, companies can improve processes and set benchmarks for quality improvement. This proactive approach can reduce manufacturing defects and increase efficiency.
Quality Control in Manufacturing
Quality Control in Manufacturing is crucial and involves measures to ensure products meet certain standards.
In the context of our problem, checking the defect rates of precision bearings is a form of quality control. This involves inspecting batches for defects, which can then inform strategies for product improvement.
  • Regular inspections help maintain high quality
  • Data-driven decisions improve production processes
Understanding probability helps in setting realistic quality control goals, such as minimizing batch rejection rates and increasing product reliability. A focus on quality control significantly impacts a company's reputation and customer satisfaction.

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Most popular questions from this chapter

A television manufacturer sold 36000 TV sets of which 297 were returned within 12 months with faults. (a) Calculate the probability that a TV set, chosen at random, is returned within 12 months. (b) A store buys 500 TV sets from the manufacturer. How many can be expected to develop faults within 12 months?

The resistance, in ohms, of a certain type of resistor is measured many times. The results are $$ \begin{array}{cc} \hline \text { Resistance }(\Omega) & \text { Frequency } \\ \hline 4.7 & 6 \\ 4.8 & 11 \\ 4.9 & 4 \\ 5.0 & 8 \\ 5.1 & 3 \\ 5.2 & 7 \\ \hline \end{array} $$ Calculate the mean value of the resistance, giving your answer to 1 d.p.

Components are made by machines \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D. Machine A makes \(30 \%\) of the components, machine B makes \(17 \%\), machine C makes \(21 \%\) and machine D makes the rest. Calculate the probability that a component is made by (a) machine A or machine B (b) machine B or machine D (c) machine A or machine B or machine D.

Table \(5.3\) shows the lifespan, \(L\), of 3958 car batteries. $$ \begin{aligned} &\text { Table } 5.3\\\ &\begin{array}{rr} \hline \text { Lifespan (months) } & \text { Number of batteries } \\ \hline L<12 & 56 \\ 12 \leq L<24 & 219 \\ 24 \leq L<36 & 436 \\ 36 \leq L<48 & 1621 \\ 48 \leq L<60 & 1319 \\ L \geq 60 & 307 \\ \hline \end{array} \end{aligned} $$ Calculate the probability that a battery picked at random (a) has a lifespan of between 24 and 36 months (b) has a lifespan of between 12 and 48 months (c) fails to work more than 48 months (d) has a lifespan of between 36 and 48 months given it is working after 24 months (e) has a lifespan greater than 60 months given it is working after 48 months (f) fails to last more than 48 months given it has lasted 36 months.

Resistors are manufactured by machines \(\mathrm{A}, \mathrm{B}\) and C. Machine A manufactures \(32 \%\) of the production, machine B manufactures \(28 \%\) of the production and machine C makes the rest. When made by machine \(A, 3 \%\) of the resistors are faulty, when made by machine B, \(4.5 \%\) are faulty, and when made by machine \(\mathrm{C}, 2.7 \%\) are faulty. A resistor is picked at random. Calculate the probability that it is (a) made by machine C (b) made by machine \(\mathrm{A}\) or machine \(\mathrm{C}\) (c) faulty, (d) faulty, given it is made by machine B (e) made by machine B, given it is faulty (f) made by machine \(\mathrm{A}\), given it is not faulty (g) not made by machine B given it is not faulty.
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