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Chapter 15: Problem 44

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

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

Step by step solution

01

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.
02

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 \( P(\text{D}) \).
03

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).
04

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 \( P(\text{D} | \text{C}) = 0.04 \).
05

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:\[ P(\text{D}) = 0.007 + 0.004 + 0.004 = 0.015 \].
06

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:\[ P(\text{A} | \text{D}) = \frac{0.007}{0.015} \approx 0.467 \].

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Key Concepts

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

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.

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Most popular questions from this chapter

In its monthly report, the local animal shelter states that it currently has 24 dogs and 18 cats available for adoption. Eight of the dogs and 6 of the cats are male Find each of the following conditional probabilities if an animal is selected at random: a) The pet is male, given that it is a cat. b) The pet is a cat, given that it is female. c) The pet is female, given that it is a dog.

The soccer team's shirts have arrived in a big box, and people just start grabbing them, looking for the right size. The box contains 4 medium, 10 large, and 6 extra-large shirts. You want a medium for you and one for your sister. Find the probability of each event described. a) The first two you grab are the wrong sizes. b) The first medium shirt you find is the third one you check. c) The first four shirts you pick are all extra-large. d) At least one of the first four shirts you check is a medium.

If you draw a card at random from a wellshuffled deck, is getting an ace independent of the suit? Explain.

Dan's Diner employs three dishwashers. Al washes \(40 \%\) of the dishes and breaks only \(1 \%\) of those he handles. Betty and Chuck each wash \(30 \%\) of the dishes, and Betty breaks only \(1 \%\) of hers, but Chuck breaks \(3 \%\) of the dishes he washes. (He, of course, will need a new job soon. ....) You go to Dan's for supper one night and hear a dish break at the sink. What's the probability that Chuck is on the job?

You draw a card at random from a standard deck of 52 cards. Find each of the following conditional probabilities: a) The card is a heart, given that it is red. b) The card is red, given that it is a heart. c) The card is an ace, given that it is red. d) The card is a queen, given that it is a face card.
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