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Chapter 2: Problem 4

Each of a sample of four home mortgages is classified as fixed rate \((F)\) or variable rate \((V)\). a. What are the 16 outcomes in \(s\) ? b. Which outcomes are in the event that exactly three of the selected mortgages are fixed rate? c. Which outcomes are in the event that all four mortgages are of the same type? d. Which outcomes are in the event that at most one of the four is a variable- rate mortgage? e. What is the union of the events in parts (c) and (d), and what is the intersection of these two events? f. What are the union and intersection of the two events in parts (b) and (c)?

Short Answer

Expert verified
a: 16 outcomes are listed. b: [FFFV, FFVF, FVFF, VFFF]. c: [FFFF, VVVV]. d: [FFFF, FFFV, FVFF, VFFF]. e: Union: [FFFF, FFFV, FVFF, VFFF, VVVV], Intersection: [FFFF]. f: Union: [FFFV, FFVF, FVFF, VFFF, FFFF, VVVV], Intersection: empty set.

Step by step solution

01

Determine Possible Outcomes

Each mortgage can either be fixed rate \((F)\) or variable rate \((V)\). For 4 mortgages, each can independently be \(F\) or \(V\). The total possible outcomes can be calculated as \(2^4 = 16\). Thus, the sample space \(s\) of outcomes is: \([FFFF, FFFV, FFVF, FFVV, FVFF, FVFV, FVVF, FVVV, VFFF, VFFV, VFVF, VFVV, VVFF, VVFV, VVVF, VVVV]\).
02

Exactly Three Fixed Rate

We need outcomes with exactly three \(F\)s and one \(V\). The valid outcomes are \([FFFV, FFVF, FVFF, VFFF]\).
03

All Mortgages Same Type

Outcomes where all mortgages are the same type include all \(F\)s or all \(V\)s. These outcomes are \([FFFF, VVVV]\).
04

At Most One Variable

Outcomes with at most one variable \((V)\) include no \(V\) or exactly one \(V\). The valid outcomes are \([FFFF, FFFV, FVFF, VFFF]\).
05

Union of Events from (c) and (d)

The union of identical mortgages and at most one variable is the set of distinct outcomes from both criteria: \([FFFF, FFFV, FVFF, VFFF, VVVV]\).
06

Intersection of Events from (c) and (d)

The intersection of the events where all mortgages are the same type and at most one is variable includes \([FFFF]\).
07

Union and Intersection of Events from (b) and (c)

The union from exact three fixed and same type mortgages is \([FFFV, FFVF, FVFF, VFFF, FFFF, VVVV]\). Their intersection is empty \([]\) since they have no common outcomes.

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Key Concepts

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

Sample Space
In probability theory, the sample space is a crucial concept used to describe all possible outcomes of an experiment.
Consider a survey of four home mortgages where each could be either a fixed rate (F) or a variable rate (V).
The sample space encompasses all the combinations of these mortgage types. Since each mortgage has two possible types and there are four mortgages, we compute the sample space as 2^4 = 16 possible outcomes.
The outcomes are written as sequences of F鈥檚 and V鈥檚, such as FFFF, FFFV, FFVF, and so forth, representing every possible combination.
This comprehensive list of 16 outcomes encapsulates the entire set of possibilities when classifying each mortgage independently.
Event Intersection
Event intersection refers to the set of outcomes that are common to two or more events.
To determine the intersection of events in probability, we focus on outcomes that satisfy all the conditions simultaneously.
In our mortgage example, consider the events where all mortgages are of the same type ([FFFF, VVVV]) and where there is at most one variable-rate mortgage ([FFFF, FFFV, FVFF, VFFF]).
The intersection of these two events yields FFFF, as it is the only outcome that fits both conditions of being the same type and having at most one variable mortgage.
Intersections help identify the overlapping elements between multiple events, offering insights into the relationships between different conditions.
Union of Events
In probability, the union of events represents all outcomes that belong to at least one of the considered events.
Unlike intersections, where all conditions must be met simultaneously, a union includes any outcome that meets at least one of the event criteria.
For instance, when combining the event of exactly three fixed-rate mortgages ([FFFV, FFVF, FVFF, VFFF]) with the event where all mortgages are the same ([FFFF, VVVV]), the union results in [FFFV, FFVF, FVFF, VFFF, FFFF, VVVV].
This set contains every outcome from both events, showcasing the comprehensive possibilities that satisfy either condition.
Understanding unions is vital for grasping the full spectrum of potential scenarios in probability.
Fixed and Variable Rate Mortgages
Fixed and variable rate mortgages are two primary types of mortgage products available to borrowers.
A fixed-rate mortgage offers a set interest rate that remains constant for the duration of the loan term. This can provide predictable payments, which is beneficial for budgeting.
In contrast, a variable-rate mortgage starts with a typically lower interest rate that can fluctuate over time based on market conditions. This offers potential savings when rates drop but carries the risk of increased payments if rates rise.
Choosing between these options depends on an individual's financial situation, risk tolerance, and market predictions.
In analyzing probability problems, understanding these terms allows for accurate classification of mortgage outcomes, such as the one in our sample space scenario.

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Most popular questions from this chapter

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