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

Based on the given information, calculate the proportion of defective items produced by the machine. Given information: 1. Probability of producing a defective item when the worker follows instructions: 0.01 2. Probability of producing a defective item when the worker does not follow instructions: 0.03 3. Probability the worker follows the instructions: 0.9 (90% of the time) Using the law of total probability, we find the proportion of defective items produced by the machine is 0.012 or 1.2%.

Step-by-step solution

Identify the given information

In this problem, we are given the following information: 1. Probability of producing a defective item when the worker follows instructions: 0.01 2. Probability of producing a defective item when the worker does not follow instructions: 0.03 3. Probability the worker follows the instructions: 0.9 (90% of the time)

Calculate the probability of producing a defective item

Let's use the law of total probability to find the probability of producing a defective item. We'll compute this by adding the probabilities of producing a defective item when the worker follows instructions and of producing a defective item when the worker does not follow instructions. Let A1 be the event that the worker follows the instructions and A2 be the event that the worker does not follow instructions. Let B be the event that the machine produces a defective item. We want to find P(B). According to the law of total probability: P(B) = P(B|A1) * P(A1) + P(B|A2) * P(A2) We know the following probabilities: P(B|A1) = 0.01 P(B|A2) = 0.03 P(A1) = 0.9 P(A2) = 1 - 0.9 = 0.1 Now, let's compute P(B): P(B) = 0.01 * 0.9 + 0.03 * 0.1

Compute the result

Now, let's calculate the value of P(B): P(B) = 0.01 * 0.9 + 0.03 * 0.1 = 0.009 + 0.003 = 0.012 So, the proportion of all items produced by the machine that will be defective is 0.012 or 1.2%.

Key concepts

Law of Total Probability

The Law of Total Probability is a fundamental rule used in probability theory to calculate the overall probability of an event based on multiple possible scenarios. Here, it's like breaking a big question into smaller parts.
In this exercise, the event of interest is producing a defective item. We have two scenarios: the worker follows instructions and the worker does not. By evaluating each scenario separately and then adding the probabilities, we determine the total probability of a defective item.
The formula used is:

• Let \( A_1 \) be the event of following instructions.
• Let \( A_2 \) be the event of not following instructions.
• Let \( B \) be the defective item produced.

So, the Law of Total Probability gives us:\[P(B) = P(B|A_1) \times P(A_1) + P(B|A_2) \times P(A_2)\] This calculation is vital as it gives us a complete picture, considering all unique pathways that lead to defects.

Defective Items

Defective items refer to those that do not meet quality standards due to faults in the production process. Understanding how often defects occur helps manage and improve production quality.
In our exercise, two factors influence whether an item is defective: the worker's compliance with machine instructions and the inherent defect rate based on compliance. When instructions are followed, the probability of defectiveness is lower compared to when they are not.
Here is what these probabilities mean:

• When instructions are followed: \( P(B|A_1) = 0.01 \)
• When instructions are not followed: \( P(B|A_2) = 0.03 \)

Having these probabilities helps identify which scenarios contribute more to defective products, allowing targeted improvements.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. In simpler terms, it's like asking, "If this condition is true, then what is the chance of that happening?"
For instance, in this problem:

• \( P(B|A_1) \) is the probability of a defect given that the worker follows instructions.
• \( P(B|A_2) \) is the probability of a defect given that the worker does not follow instructions.

Conditional probabilities help isolate different variables' impacts. They inform decisions about where controls or changes could yield improvements, especially in processes like production.

Probability Calculation

Probability calculation enables us to quantify uncertainty. It involves determining the likelihood of various outcomes.
In our exercise, the main calculation performed is:\[P(B) = 0.01 \times 0.9 + 0.03 \times 0.1\]Breaking down this calculation:

• First term: \( 0.01 \times 0.9 \) = 0.009, which accounts for defects when instructions are followed.
• Second term: \( 0.03 \times 0.1 \) = 0.003, which accounts for defects when instructions are not followed.

Adding these gives \( 0.012 \) or 1.2%. This calculation shows how often defects occur under different conditions and their combined impact on the overall defect rate.