Question

A company manufacturing electronic components for home entertainment systems buys electrical connectors from three suppliers. The company prefers to use supplier A because only \(1 \%\) of those connectors prove to be defective, but supplier A can deliver only \(70 \%\) of the connectors needed. The company must also purchase connectors from two other suppliers, \(20 \%\) from supplien \(B\) and the rest from supplier \(\mathrm{C}\). The rates of defective connectors from \(\mathrm{B}\) and \(\mathrm{C}\) are \(2 \%\) and \(4 \%\), respectively. You buy one of these components, and when you try to use it you find that the connector is defective. What's the probability that your component came from supplier A?

Short answer

The probability that the component came from supplier A is approximately 46.7%.

Step-by-step solution

Define the Events

Let:- \( \text{A} \) be the event that the component came from supplier A.- \( \text{D} \) be the event that the component is defective.We need to find \( P(\text{A} | \text{D}) \), the probability that the component came from supplier A given that it is defective.

Use Bayes' Theorem

Bayes' theorem states, \[ P(\text{A} | \text{D}) = \frac{P(\text{D} | \text{A}) P(\text{A})}{P(\text{D})} \]. We need to find \( P(\text{D} | \text{A}) \), \( P(\text{A}) \), and \( P(\text{D}) \).

Calculate the Probabilities for Supplier A

\(- P(\text{A}) = 0.7 \) (since 70% of the components are from supplier A).- \( P(\text{D} | \text{A}) = 0.01 \) (since 1% of components from supplier A are defective).

Calculate the Total Probability of Defect

The probability \( P(\text{D}) \) can be calculated using the law of total probability:\[ P(\text{D}) = P(\text{D} | \text{A})P(\text{A}) + P(\text{D} | \text{B})P(\text{B}) + P(\text{D} | \text{C})P(\text{C}) \].Given, \( P(\text{B}) = 0.2 \), \( P(\text{C}) = 0.1 \), \( P(\text{D} | \text{B}) = 0.02 \), and \( P(\text{D} | \text{C}) = 0.04 \).

Plug in Values to the Total Probability

Calculate \( P(\text{D}) \) using the values:\[ P(\text{D}) = (0.01)(0.7) + (0.02)(0.2) + (0.04)(0.1) \].Simplifying gives:\[ P(\text{D}) = 0.007 + 0.004 + 0.004 = 0.015 \].

Apply Values to Bayes' Theorem

Now, substituting these values back into Bayes' theorem:\[ P(\text{A} | \text{D}) = \frac{(0.01)(0.7)}{0.015} \].Simplifying gives:\[ P(\text{A} | \text{D}) = \frac{0.007}{0.015} \approx 0.467 \].

Key concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. This concept is central to many probability problems, including those involving Bayes' Theorem. In our exercise, we need to find the probability of buying a defective connector from supplier A after knowing the connector is defective.

In mathematical terms, this is denoted as \( P(A | D) \), the probability that a component comes from supplier A given it is defective. To understand this better, imagine we only consider the subset of connectors that are defective. Conditional probability allows us to focus on this subset and reevaluate odds within it.

The expression for conditional probability is:

• \( P(A | D) = \frac{P(A \cap D)}{P(D)} \)

Where \( P(A \cap D) \) is the probability that both events A and D happen. In simpler words, it's about how likely it is to pick something from supplier A, knowing it's defective.

Law of Total Probability

The Law of Total Probability helps us find the probability of an event based on probabilities conditioned on another set of events. It is fundamental when calculating the total probability of a complex event by considering multiple mutually exclusive scenarios that can lead to that event.

In the given exercise, we used the Law of Total Probability to find the overall probability that a connector is defective (event D), considering the connectors might come from one of three different suppliers. Here's how it works:

• First, partition all events into distinct cases that cover all possibilities. For this problem, that's whether a connector comes from A, B, or C.
• Calculate the probability of the unwanted event (defective) for each subset.
• The total probability of defect \( P(D) \) is given by: \[ P(D) = P(D | A)P(A) + P(D | B)P(B) + P(D | C)P(C) \]

This formula ensures that we account for all possible ways to achieve a defective connector, considering different defective rates and proportions from each supplier.

Defective Rate

Defective rate refers to the proportion of defective items produced or supplied by a provider. In our study case, different suppliers each have their own defective rates for the connectors they supply:

• Supplier A has a defective rate of 1%.
• Supplier B has a defective rate of 2%.
• Supplier C has a defective rate of 4%.

Understanding defective rate is crucial as it affects the probability of a component being defective based on its source.

When dealing with multiple suppliers, each with a unique defective rate, evaluating how these rates contribute to the overall defectiveness helps us make informed decisions about sourcing. It also helps in applying theories like Bayes' to better predict the source of defects in a mixed supply.

In our example, lower defective rates from supplier A make it more desirable, highlighting how defective rates impact supplier preference and decision-making.