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Chapter 2: Problem 30

A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches: (a) The batch contains the items \(\\{a, b, c, d\\}\). (b) The batch contains the items \(\\{a, b, c, d, e, f, g\\}\). (c) The batch contains 4 defective items and 20 good items. (d) The batch contains 1 defective item and 20 good items.

Short Answer

Expert verified
The ordered sample spaces for each part are: (a) 12 pairs, (b) 42 pairs, (c) combinations of D and G, (d) 40 pairs.

Step by step solution

01

Define Sample Space for Batch (a)

For batch (a) with items \(\{a, b, c, d\}\), we need to form ordered pairs of samples without replacement. This means each pair lists the selected items in order, and each item can only be selected once. The ordered sample space is: \((a, b), (a, c), (a, d), (b, a), (b, c), (b, d), (c, a), (c, b), (c, d), (d, a), (d, b), (d, c)\).
02

Define Sample Space for Batch (b)

For batch (b) with items \(\{a, b, c, d, e, f, g\}\), we again list all ordered pairs where no item repeats. The sample space includes 42 combinations: \((a, b), (a, c), (a, d), (a, e), (a, f), (a, g), (b, a), (b, c), (b, d), (b, e), (b, f), (b, g), (c, a), (c, b), (c, d), (c, e), (c, f), (c, g), (d, a), (d, b), (d, c), (d, e), (d, f), (d, g), (e, a), (e, b), (e, c), (e, d), (e, f), (e, g), (f, a), (f, b), (f, c), (f, d), (f, e), (f, g), (g, a), (g, b), (g, c), (g, d), (g, e), (g, f)\).
03

Define Sample Space for Batch (c)

In the case where we have 4 defective (D) and 20 good (G) items in batch (c), we use D for defective and G for good. The ordered sample space is formed by creating pairs such as \((D_1, G_1), (D_1, G_2), ..., (D_1, D_2), (G_1, D_2),...\) considering each specific combination of these items, while ensuring no repeats within a pair. The exhaustive list requires considering all combinations of picking first a D or G and then another item.
04

Define Sample Space for Batch (d)

Finally, for batch (d) with 1 defective and 20 good items, there are only two types of pairs: one where the defective item is first, and the remaining is a good-item pair: \((D, G_1), (D, G_2), ..., (D, G_{20})\) and the other where a good item is selected first, followed by the defective item: \((G_1, D), (G_2, D), ..., (G_{20}, D)\).

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Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics that deals with the likelihood of various outcomes. It helps us quantify uncertainty and predict the occurrence of events. In our exercise, this involves understanding the sample space, which is crucial for calculating probabilities. A sample space is the set of all possible outcomes of a probability experiment. For instance, when dealing with batches, each selection of items represents one outcome from which probability can be calculated. By systematically listing these possibilities, you can apply probability rules to determine the likelihood of choosing certain combinations.
Combinatorics
Combinatorics is all about counting and arranging items. It's a fundamental aspect of probability because it helps us figure out how many possible ways we can arrange or select items from a set. Often, combinatorics is concerned with finding all possible combinations and permutations in a given scenario. In the context of selecting items from a batch, combinatorics assists in listing all ordered pairs, telling us exactly how many different sample possibilities there are. By understanding combinatorics, one can determine the sample size and thus proceed with calculating probabilities.
Ordered Sampling
Ordered sampling is the process of selecting a number of items from a set in such a way that the order in which you select them matters. From our exercise, ordered sampling without replacement means that once an item is selected, it can't be chosen again, and the sequence in which items are picked changes the result. In both batches (a) and (b) of the exercise, the sequence (a, b) is different from (b, a) because the order is significant. Listing ordered samples is important when the outcome itself depends on the arrangement, influencing probability distributions.
Defective and Good Items
The distinction between defective and good items is a practical consideration in many probability problems. Here, items are classified into two types: defective (likely undesirable) and good (preferable). In scenarios like batches (c) and (d), this classification alters the sample space formation because it changes the nature and the count of each type available. For instance, in a batch with 4 defective and 20 good items, numerous combinations exist for selecting one defective and one good. The concept of defective and good items helps in assessing risks and probabilities associated with quality control and inventory management.

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Most popular questions from this chapter

Four bits are transmitted over a digital communications channel. Each bit is either distorted or received without distortion. Let \(A_{i}\) denote the event that the \(i\) th bit is distorted, \(i=1, \ldots, 4\) (a) Describe the sample space for this experiment. (b) Are the \(A_{i}\) 's mutually exclusive? Describe the outcomes in each of the following events: (c) \(A_{1}\) (d) \(A_{1}^{\prime}\) (e) \(A_{1} \cap A_{2} \cap A_{3} \cap A_{4}\) (f) \(\left(A_{1} \cap A_{2}\right) \cup\left(A_{3} \cap A_{4}\right)\)

In control replication, cells are replicated over a period of two days. Not until mitosis is completed can freshly synthesized DNA be replicated again. Two control mechanisms have been identified - one positive and one negative. Suppose that a replication is observed in three cells. Let \(A\) denote the event that all cells are identified as positive and let \(B\) denote the event that all cells are negative. Describe the sample space graphically and display each of the following events: (a) \(A\) (b) \(B\) (c) \(A \cap B\) (d) \(A \cup B\)

A lot of 100 semiconductor chips contains 20 that are defective. (a) Two are selected, at random, without replacement, from the lot. Determine the probability that the second chip selected is defective. (b) Three are selected, at random, without replacement, from the lot. Determine the probability that all are defective.

A batch of 350 samples of rejuvenated mitochondria contains eight that are mutated (or defective). Two are selected, at random, without replacement from the batch. (a) What is the probability that the second one selected is defective given that the first one was defective? (b) What is the probability that both are defective? (c) What is the probability that both are acceptable?

A batch of 500 containers for frozen orange juice contains five that are defective. Two are selected, at random, without replacement from the batch. (a) What is the probability that the second one selected is defective given that the first one was defective? (b) What is the probability that both are defective? (c) What is the probability that both are acceptable? Three containers are selected, at random, without replacement, from the batch. (d) What is the probability that the third one selected is defective given that the first and second ones selected were defective? (e) What is the probability that the third one selected is defective given that the first one selected was defective and the second one selected was okay? (f) What is the probability that all three are defective?
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