Question

Components of a certain type are shipped to a supplier in batches of \(10 .\) Suppose that \(50 \%\) of all batches contain no defective components, \(30 \%\) contain one defective component, and \(20 \%\) contain two defective components. A batch is selected at random. Two components from this batch are randomly selected and tested. a. If the batch from which the components were selected actually contains two defective components, what is the probability that neither of these is selected for testing? b. What is the probability that the batch contains two defective components and that neither of these is selected for testing? c. What is the probability that neither component selected for testing is defective? (Hint: This could happen with any one of the three types of batches. A tree diagram might help.)

Short answer

a. The probability neither component is defective if there are two defective ones in the batch is \( 28/45 \), \n b. The probability that the batch contains two defective components and neither is defective is \(0.2 * 28/45\), \n c. The probability that neither component selected for testing is defective is \(0.5*1 + 0.3*(C(9,2)/C(10,2)) + 0.2*(C(8,2)/C(10,2))\).

Step-by-step solution

Probability neither component is defective if there are two in the batch

We determine the probability neither component is defective if there are two defective ones in the batch using combinations. We have two defective components, meaning there are 8 working ones. Choosing 2 out of those 8 without replacement gives us \(C(8,2) = 28\) ways. There are total of \(C(10,2) = 45\) ways we can choose 2 components from the batch. Hence the probability will be the ratio of those two, which equates to \(P = 28/45\).

Probability batch contains two defective components but neither is chosen

To determine the probability that the batch contains two defective components and neither is tested, we must consider the probability of selecting a batch that contains exactly two defective components. We know that this is 20 percent, or 0.2. Hence, the desired probability is product of the probability of selecting a batch that contain two defective components and neither is chosen for testing, which is \(0.2 * (28/45)\).

Calculate probability that neither component selected for testing is defective

Here we have three types of batches, one with 0 defective components, one with 1 defective component, and lastly one with two defective components. The probabilities of choosing a non-defective component from each of these is respectively \(1, C(9,2) / C(10,2), C(8,2) / C(10,2)\), and the probabilities of these batches are 0.5, 0.3, and 0.2. So, the total probability will be a sum of each of these probabilities multiplied by corresponding batch probability. That gives us the result: \(0.5*1 + 0.3*(C(9,2)/C(10,2)) + 0.2*(C(8,2)/C(10,2))\).

Key concepts

Probability with Defective Components

Understanding the probability of encountering defective components in a batch is crucial in quality control and risk management. Let's imagine a scenario where you have multiple batches of products, and some contain defective items. The probability of selecting a defective or non-defective component depends on the composition of the batch you're dealing with.

For instance, if you know a certain batch contains two defective components out of ten, you could ask: What are the odds of randomly picking two items and both being non-defective? This problem requires combinatorial probability knowledge to solve, which involves calculating the number of favorable outcomes over all possible outcomes. Specifically, if you have eight non-defective components, there are C(8,2) = 28 ways to choose two non-defective components out of the possible C(10,2) = 45 ways to choose any two components.

With these combinations, you can compute the probability by dividing the number of favorable outcomes (selecting non-defective components) by the total number of possible outcomes (selecting any two components). So, if you aim to improve your proficiency in dealing with defective components in statistics, grasping the foundations of combinatorial calculations and thinking critically about batch compositions are essential steps.

Combinatorial Probability

Combinatorial probability is a field of mathematics that deals with counting and calculating the likelihood of events occurring, based on the combination of various outcomes. In our example, we use combinations to determine the number of ways we can select two non-defective components from a batch.

Combinations are a specific way to calculate how many different groups can be formed from a larger set of items, where the order of the items does not matter. The formula for calculating the combination of choosing k items from a set of n is defined as C(n,k) = n! / (k!(n-k)!), where ! denotes factorial, the product of all positive integers up to that number.

Real-World Applications

Combinatorial probability isn't just theoretical; it finds applications in various fields such as genetics, marketing, finance, and more. For instance, it helps geneticists calculate the odds of inheriting specific traits, while marketers may use it to predict the potential combinations consumers might choose when selecting product features.

Tree Diagram in Probability

A tree diagram is a visual representation that helps to map out all possible outcomes of a probability event. It's like a branching diagram that shows all possible paths from start to finish, where each branch represents a possible decision or chance event.

Using a tree diagram is especially useful when dealing with multiple stages of probability events. In our textbook exercise, for instance, to calculate the probability that neither component selected for testing is defective, we can use a tree diagram to visualize the three different types of batches (with 0, 1, or 2 defective components).

Each branch of the tree diagram would represent the probabilities of selecting each batch type, and further branches would represent the probabilities of selecting non-defective components from those batches. By multiplying along the branches and adding the probabilities of each 'end' branch, we can determine the overall probability of selecting non-defective components. Tree diagrams provide an intuitive way to understand complex probability questions and can be an invaluable tool for students to grasp multi-stage probability events.