91Ó°ÊÓ

Based on the following table, which shows the performance of a selection of 100 stocks after one year. (Take S to be the set of all stocks represented in the table.) $$ \begin{array}{|r|c|c|c|c|} \hline & \multicolumn{3}{|c|} {\text { Companies }} & \\ \cline { 2 - 4 } & \begin{array}{c} \text { Pharmaceutical } \\ \boldsymbol{P} \end{array} & \begin{array}{c} \text { Electronic } \\ \boldsymbol{E} \end{array} & \begin{array}{c} \text { Internet } \\ \boldsymbol{I} \end{array} & \text { Total } \\ \hline \begin{array}{r} \text { Increased } \\ \boldsymbol{V} \end{array} & 10 & 5 & 15 & 30 \\ \hline \begin{array}{r} \text { Unchanged }^{*} \\ \boldsymbol{N} \end{array} & 30 & 0 & 10 & 40 \\ \hline \begin{array}{r} \text { Decreased } \\ \boldsymbol{D} \end{array} & 10 & 5 & 15 & 30 \\ \hline \text { Total } & 50 & 10 & 40 & 100 \\ \hline \end{array} $$ If a stock stayed within \(20 \%\) of its original value, it is classified as "unchanged." Find all pairs of events that are not mutually exclusive among the events \(P, E, I, V, N\), and \(D .\)

Step by step solution

01

Understand the given information

The given table shows the performance of 100 stocks in three different sectors: Pharmaceutical (P), Electronic (E), and Internet (I). The performance is classified into three categories: Increased (V), Unchanged (N), and Decreased (D). Step 2: Find possible overlaps

02

Identify pairs of events that can occur simultaneously

Now, we will find all pairs of events that are not mutually exclusive; these are pairs of events that can happen at the same time. We will do this by considering each pair of events and searching for any overlaps in the table. 1. P and V: A stock can be both Pharmaceutical and have Increased value. 2. P and N: A stock can be both Pharmaceutical and have Unchanged value. 3. P and D: A stock can be both Pharmaceutical and have Decreased value. 4. E and V: A stock can be both Electronic and have Increased value. 5. E and N: A stock can be both Electronic and have Unchanged value 6. E and D: A stock can be both Electronic and have Decreased value. 7. I and V: A stock can be both Internet and have Increased value. 8. I and N: A stock can be both Internet and have Unchanged value. 9. I and D: A stock can be both Internet and have Decreased value. Step 3: Conclusion

03

List all pairs of events that are not mutually exclusive

From the analysis of the table, we find that the following pairs of events are not mutually exclusive: 1. P and V 2. P and N 3. P and D 4. E and V 5. E and N 6. E and D 7. I and V 8. I and N 9. I and 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 different outcomes occurring. When we talk about events in probability, we're referring to outcomes or occurrences that can happen in a given scenario. One key concept in this area is the idea of mutually exclusive events.

Mutually exclusive events are events that cannot both happen at the same time. For example, when flipping a coin, the events 'heads' and 'tails' are mutually exclusive because the coin cannot land on both sides at once. In contrast, non-mutually exclusive events can occur together, such as drawing a card that is both red and a king from a deck of standard playing cards.

In the context of the stock performance analysis from the exercise, non-mutually exclusive events are identified by looking at events that represent categories (like Pharmaceutical, Electronic, Internet) and outcomes (Increased, Unchanged, Decreased) that can coincide. A practical example would be a Pharmaceutical stock that has Increased in value, which is both a category and an outcome happening at the same time.

Stock Performance Analysis

Analyzing stock performance involves examining how a stock's value has changed over time. This can include looking at whether the stock's price has increased, remained relatively unchanged, or decreased. Investors and analysts use this information to make predictions about future performance and to decide where to allocate their resources.

In the provided exercise, we examine a selection of stocks categorized by sector and their performance over a year. Understanding how to analyze this information is crucial. Knowing that the events of increased value ('V'), unchanged value ('N'), and decreased value ('D') can occur across different sectors can inform investors' decisions. For instance, if the Pharmaceuticals sector has many stocks that increased in value, this could indicate a robust sector performance and may influence investment strategies.

Sector-wise Stock Categorization

Sector-wise stock categorization is the process of grouping stocks based on their respective sectors or industries, such as Pharmaceutical, Electronic, and Internet, as seen in the exercise. This is an important practice in finance because it allows investors to diversify their portfolios and manage risk more effectively by not being overly exposed to the fluctuations of a single sector.

When investors understand sector-wise categorization, they can track performance trends within each sector. This categorization aids in identifying which sectors are outperforming or underperforming. In our exercise, a table presents the number of stocks in each sector along with their stock performance over one year. This type of data can signal industry trends and inform investment choices, as certain sectors may show more stability or growth potential than others.