91Ó°ÊÓ

In probability and statistics, the concept of a sample space is fundamental. The sample space, denoted as \( \mathcal{S} \), refers to the set of all possible outcomes of a particular experiment. In our case, the experiment consists of classifying four home mortgages as either fixed rate \((F)\) or variable rate \((V)\). Hence, each mortgage labeled as either \( F \) or \( V \) gives rise to a different combination of outcomes.Since each mortgage has two classification possibilities, the total number of possible combinations for four mortgages is calculated by raising two to the fourth power, \( 2^4 \). This means there are 16 possible outcomes, each one representing a unique sequence of \( F \) and \( V \) decisions:

• \( \text{FFFF} \)
• \( \text{FFFV} \)
• \( \text{FFVF} \)
• \( \text{FFVV} \)
• \( \text{FVFF} \)
• \( \text{FVFV} \)
• \( \text{FVVF} \)
• \( \text{FVVV} \)
• \( \text{VFFF} \)
• \( \text{VFFV} \)
• \( \text{VFVF} \)
• \( \text{VFVV} \)
• \( \text{VVFF} \)
• \( \text{VVFV} \)
• \( \text{VVVF} \)
• \( \text{VVVV} \).

This complete list of outcomes is the sample space for the experiment.

Events in probability and statistics are specific outcomes or sets of outcomes from the sample space that hold particular interest. They are subsets of the sample space \( \mathcal{S} \), highlighting the diversity of possible results. In our exercise, several important events have been identified. Let's break these down to understand better.

Exactly Three Fixed-Rate Mortgages

This event captures the sequences where three out of the four mortgages are a fixed-rate \( F \). Through careful examination, the outcomes that meet this condition are \( \text{FFFV} \), \( \text{FFVF} \), \( \text{FVFF} \), and \( \text{VFFF} \). Each sequence, although differently arranged, includes three \( F \)'s, highlighting the commonality within this event.

Same Type Mortgages

Another event of interest is when all mortgages are classified under the same type. This limits the possible outcomes to those that are uniformly \( F \) or \( V \). These outcomes are \( \text{FFFF} \) and \( \text{VVVV} \).

At Most One Variable-Rate Mortgage

When looking at mortgages having at most one variable rate, we are interested in outcomes with zero or one \( V \). The viable outcomes here are \( \text{FFFF} \) (zero \( V \)) and \( \text{FFFV} \) (one \( V \)). These subsets provide insights into specific elements of the sample space, each focusing on different characteristic configurations of mortgages.

Discrete probability deals with scenarios where sample spaces consist of a finite number of outcomes, making it suitable for analysis in this context. When examining discrete probability, each outcome in a sample space has a specific probability assigned to it. These probabilities add up to one.Let's explore how discrete probability applies to some events in this exercise:

Union and Intersection of Events

The union of two events includes any outcome that belongs to either event or both. In our scenario:- For events where all mortgages are of the same type \((\text{FFFF, VVVV})\) and where at most one is a variable rate \((\text{FFFF, FFFV})\), the union combines to result in outcomes \( \text{FFFF, FFFV, VVVV} \).The intersection of two events, however, involves outcomes common to both events. In the same example, the intersection results in \( \text{FFFF} \) as the only outcome shared by both events.

Probabilistic Measures

Since each of the 16 outcomes in our sample space has an equal chance of occurring, the probability of any single outcome is \( \frac{1}{16} \). To find the probability of an event:

• Count the number of outcomes that belong to the event.
• Divide by the total number of outcomes in the sample space.

For instance, the probability of selecting mortgages where exactly three are fixed-rate \((\text{FFFV, FFVF, FVFF, VFFF})\) is \( \frac{4}{16} = \frac{1}{4} \). This discrete approach facilitates clear and calculable understanding of probabilities in such experiments.