Question

A worker-operated machine produces a defective item with probability .01 if the worker follows the machine's operating instructions exactly, and with probability .03 if he does not. If the worker follows the instructions \(90 \%\) of the time, what proportion of all items produced by the machine will be defective?

Short answer

Answer: The overall probability of producing a defective item by the machine is 0.012 or 1.2%.

Step-by-step solution

Identify the probability of the worker following instructions

The worker follows the instructions 90% of the time, so the probability of the worker following instructions is 0.9.

Identify the probability of the worker not following instructions

If the worker follows the instructions 90% of the time, that means he or she does not follow instructions 10% of the time. So, the probability of the worker not following instructions is 0.1.

Calculate the probability of producing a defective item when the worker follows instructions

Since the worker produces a defective item with a probability of 0.01 when following instructions, we multiply the probability of this happening by the probability of following instructions: 0.01 * 0.9 = 0.009

Calculate the probability of producing a defective item when the worker does not follow instructions

Similarly, the worker produces a defective item with a probability of 0.03 when not following instructions. So we multiply this probability by the probability of not following instructions: 0.03 * 0.1 = 0.003

Calculate the total probability of producing a defective item

Now, we add the probabilities calculated in steps 3 and 4 to find the overall probability of a defective item being produced by the machine: 0.009 + 0.003 = 0.012 Thus, the proportion of all items produced by the machine that will be defective is 0.012 or 1.2%.

Key concepts

Probability Theory

Probability theory is the mathematical framework that allows us to analyze random events and make predictions about the likelihood of various outcomes. At its core, it involves calculating the chances of specific events occurring, given a set of possible events. This is crucial in real-world scenarios, such as manufacturing, where companies need to assess the risk of defects.

In the context of the provided exercise, the probability theory comes into play when determining the likelihood of producing a defective item. The calculations provided are fundamental applications of this theory, where probabilities are expressed as numbers between 0 and 1, with 0 indicating an impossibility, and 1 signifying certainty.

To enhance understanding for students grappling with probability theory, it would be beneficial to relate these abstract concepts to tangible, everyday examples. Visual aids, such as probability trees or Venn diagrams, can also be used to represent the various outcomes and their probabilities in a more digestible format.

Quality Control Statistics

Quality control statistics are employed to monitor and manage the quality of products and services. It involves the collection and analysis of data to evaluate a company's production processes and outputs. In terms of quality control, the probability of defects is a critical statistic as it can directly impact customer satisfaction and a company’s reputation.

The exercise showcases a basic quality control statistic by determining the proportion of defective items produced. This is a real-world application of statistical methods in monitoring product quality and can help in identifying points of process improvements. To make this relevant to students, they should be made aware of how such statistics aid in decisions like whether a production process needs adjustment or if additional training for workers is warranted, thus connecting classroom concepts with industry practices.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. This is pivotal in situations where the likelihood of an event is influenced by the occurrence of a previous event. In our exercise, for instance, the probability of a defect changes based on whether the worker follows the machine’s operating instructions.

For clarification, imagine two intertwined events: 'A worker following instructions' and 'Producing a defective item'. Conditional probability allows us to calculate the likelihood of the second event given the first one is true. A way to improve comprehension here could be through interactive scenarios where students can manipulate the conditions to see how the overall probability changes, thereby offering a deeper understanding of the interplay between different factors.