Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A halogen desk lamp produced by Luminar was found to be defective. The company has three factories where the lamps are manufactured. The percentage of the total number of halogen desk lamps produced by each factory and the probability that a lamp manufactured by that factory is defective are shown in the accompanying table. What is the probability that the defective lamp was manufactured in factory III? $$\begin{array}{ccc}\hline & & \text { Percent of } & \text { Probability of } \\\\\text { Factory } & \text { Total Production } & \begin{array}{c}\text { Defective } \\\\\text { Component }\end{array} \\ \hline \text { I } & 35 & .015 \\\\\hline \text { II } & 35 & .01 \\ \hline \text { III } & 30 & .02 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The probability that the defective lamp was manufactured in Factory III is approximately 58.54%.

Step 1: Break down the given information

From the table, we extract the production percentages and defect probabilities for each factory. Factory I: - Production percentage: 35% - Defective probability: 0.015 Factory II: - Production percentage: 35% - Defective probability: 0.01 Factory III: - Production percentage: 30% - Defective probability: 0.02 We want to find the probability of a defective lamp being produced in Factory III, given that the lamp is defective. We denote this as P(Factory III | Defective).

Step 2: Apply Bayes' theorem

Bayes' theorem is given as: $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$ In our case, A corresponds to Factory III, and B corresponds to a defective lamp. So we rewrite Bayes' theorem as: $$P(\text{Factory III | Defective}) = \frac{P(\text{Defective | Factory III}) P(\text{Factory III})}{P(\text{Defective})}$$

Step 3: Calculate the required probabilities

We already have the following probabilities: - \(P(\text{Defective | Factory III}) = 0.02\) - \(P(\text{Factory III}) = 0.30\) Now, let's calculate the probability of a defective lamp, P(Defective). We can do this by using the Law of Total Probability, given by: $$P(\text{Defective}) = P(\text{Defective | Factory I})P(\text{Factory I}) + P(\text{Defective | Factory II})P(\text{Factory II}) + P(\text{Defective | Factory III})P(\text{Factory III})$$ So, we have: $$P(\text{Defective}) = (0.015)(0.35) + (0.01)(0.35) + (0.02)(0.30)$$

Step 4: Calculate the final probability

Now, substitute the known values into Bayes' theorem: $$P(\text{Factory III | Defective}) = \frac{(0.02)(0.30)}{(0.015)(0.35) + (0.01)(0.35) + (0.02)(0.30)} = \frac{0.006}{0.01025} \approx 0.5854$$ So, there is approximately a 58.54% chance that the defective lamp was produced in Factory III.

Key Concepts

Law of Total Probability

The Law of Total Probability is a fundamental rule used to connect probabilities to conditional probabilities. It allows us to compute the likelihood of an event when there are several different ways that the event might occur. This law is especially useful when dealing with multiple scenarios or groups, like in the case of factories producing lamps.
In simple terms, the law provides a way to break down complex problems. Imagine you want to find out the probability of defective lamps. Each factory contributes differently to the total production. By multiplying the probability of a defect in a factory by the proportion of lamps that factory produces, and then summing over all three factories, we get the total probability of a defective lamp.
In our exercise, this is shown as:

• \[P(\text{Defective}) = P(\text{Defective | Factory I}) \times P(\text{Factory I}) + P(\text{Defective | Factory II}) \times P(\text{Factory II}) + P(\text{Defective | Factory III}) \times P(\text{Factory III})\]

This method simplifies understanding and calculation by segmenting the total probability into a sum of specific instances.

Probability Calculations

Probability calculations are a way to measure how likely an event is to happen. They involve mathematical computation based on known probabilities and can be quite straightforward once you grasp the basics. Probability values range between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
To compute probabilities, we often rely on given data and known formulas. For example, in the lamp exercise, the known percentages of production and probabilities of defects are key facts used in calculations.
Calculations centers around finding precise probabilities. This involves using methods and known data effectively. In our scenario, for a defective lamp, probabilities of defect from each factory and their respective production shares were multiplied to find the overall probability of a defective lamp.
To find the final requirement using Bayes' theorem, computing \(P(\text{Factory III | Defective})\), first calculating \(P(\text{Defective})\) was crucial. Each probability calculation step helps in building up to the final solution with precision.

Conditional Probability

Conditional probability is finding the probability of an event, given that another event has happened. This type of probability is crucial when dealing with scenarios that rely on certain conditions.
Conditional probability is denoted as \(P(A|B)\), where A is the event of interest and B is the known condition. In our problem, the interest lay in determining what factory a defective lamp came from, knowing it is defective.
The step-by-step solution uses Bayes' theorem, which is built upon conditional probability, to solve for \(P(\text{Factory III | Defective})\). Bayes' theorem becomes instrumental when trying to reverse probabilities, helping to determine the conditional probability from known information.

• The formula is: \[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\], where \(A\) is the event of a lamp being from Factory III, and \(B\) is that the lamp is defective.
• This calculation involved plugging in probabilities from the total probability calculations, showing a perfect bridge between these two concepts.

Understanding conditional probability allows solving more strategic queries by considering conditions or evidence already presented.