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Chapter 4: Problem 47

A financial manager made two new investments-one in the oil industry and one in municipal bonds. After a one-year period, each of the investments will be classified as either successful or unsuccessful. Consider the making of the two investments as an experiment. a. How many sample points exist for this experiment? b. Show a tree diagram and list the sample points. c. Let \(O=\) the event that the oil industry investment is successful and \(M=\) the event that the municipal bond investment is successful. List the sample points in \(O\) and in \(M\) d. List the sample points in the union of the events \((O \cup M)\) e. List the sample points in the intersection of the events \((O \cap M)\) f. Are events \(O\) and \(M\) mutually exclusive? Explain.

Short Answer

Expert verified
a. 4 sample points exist. b. (success, success), (success, unsuccessful), (unsuccessful, success), (unsuccessful, unsuccessful). c. O: (success, success), (success, unsuccessful), M: (success, success), (unsuccessful, success). d. (success, success), (success, unsuccessful), (unsuccessful, success). e. (success, success). f. No, O and M are not mutually exclusive.

Step by step solution

01

Determine the Sample Space

For each investment (oil and municipal bonds), there are two possible outcomes: successful or unsuccessful. Hence, we have the following sample space: \[(S_s, M_s), (S_s, M_u), (S_u, M_s), (S_u, M_u)\] where \(S_s\) represents a successful oil investment, \(S_u\) represents an unsuccessful oil investment, \(M_s\) represents a successful municipal bond investment, and \(M_u\) represents an unsuccessful municipal bond investment.
02

Count the Sample Points

Each investment can be successful or unsuccessful independently. Thus, the number of sample points is the product of the possible outcomes for each: \(2 \times 2 = 4\).
03

Draw and Interpret Tree Diagram

We can use a tree diagram to illustrate each combination of investment outcomes: - Start with either a successful (S) or unsuccessful (U) oil investment.- From each outcome, branch into successful (S) or unsuccessful (U) municipal bond investment. - First branch: Oil (S) 鈫� Bonds (S, U) - Second branch: Oil (U) 鈫� Bonds (S, U)The outcomes are: \((S_s, M_s), (S_s, M_u), (S_u, M_s), (S_u, M_u)\).
04

Define Event Set O and M

Event \(O\) means the oil investment is successful: \((S_s, M_s), (S_s, M_u)\).Event \(M\) means the municipal bond investment is successful: \((S_s, M_s), (S_u, M_s)\).
05

List Sample Points in Union O 鈭� M

The union of \(O\) and \(M\) includes sample points where either the oil investment or municipal investment is successful. Hence, the union contains: \((S_s, M_s), (S_s, M_u), (S_u, M_s)\).
06

List Sample Points in Intersection O 鈭� M

The intersection of \(O\) and \(M\) contains sample points where both investments were successful. Thus, \((S_s, M_s)\) is in the intersection.
07

Determine if O and M are Mutually Exclusive

Events \(O\) and \(M\) are not mutually exclusive because \(O\) and \(M\) can both occur simultaneously; they share the sample point \((S_s, M_s)\).

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Key Concepts

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

Sample Space
The sample space is a fundamental concept in probability, representing all possible outcomes of an experiment. In the context of business decisions like investments, determining the sample space allows you to understand the range of outcomes possible from your actions.
For the given exercise, the financial manager's experiment involves two investments with two possible outcomes each: success or failure. This means each investment can independently succeed or fail, leading to a combined number of scenarios.
With two possibilities for each investment, we calculate the total number of outcomes by multiplying the possibilities: \(2 \times 2 = 4\). Consequently, the sample space for this experiment includes four potential outcomes:
  • \((S_s, M_s)\) - Both investments are successful.
  • \((S_s, M_u)\) - Oil successful, bonds not.
  • \((S_u, M_s)\) - Oil not successful, bonds successful.
  • \((S_u, M_u)\) - Both investments are unsuccessful.
Understanding the sample space lays the groundwork for calculating probabilities related to the success or failure of each investment.
Tree Diagram
A tree diagram is a visual tool used to map out all possible outcomes of an experiment systematically. It helps in understanding the structure and sequence of events and their respective probabilities, making complex scenarios easier to analyze.
In this exercise, the tree diagram begins with the first decision point: whether the oil investment is successful (S) or unsuccessful (U).
From each of these nodes, the second decision is represented: whether the municipal bond investment is successful (S) or unsuccessful (U).
This step-by-step branching leads to the sample space we've discussed:
  • From Oil (S):
    • Bonds (S) 鈫� \((S_s, M_s)\)
    • Bonds (U) 鈫� \((S_s, M_u)\)
  • From Oil (U):
    • Bonds (S) 鈫� \((S_u, M_s)\)
    • Bonds (U) 鈫� \((S_u, M_u)\)
The tree diagram effectively shows all combinations of outcomes in a visually manageable way, helping decision-makers foresee all potential paths and outcomes.
Mutually Exclusive Events
In probability, mutually exclusive events are outcomes that cannot occur simultaneously. If one event happens, the other cannot, and vice versa.
For example, tossing a coin results in either heads or tails, but never both 鈥� these are mutually exclusive outcomes.
In the context of the financial manager's investment analysis, the two events are:
  • Event \(O\): Oil industry investment is successful.
  • Event \(M\): Municipal bond investment is successful.
These events are not mutually exclusive because there exists a scenario where both can occur together, specifically for the sample point\((S_s, M_s)\), where both investments succeed.
Recognizing whether events are mutually exclusive helps in calculating probabilities correctly and understanding dependencies between outcomes.
Investment Outcomes
Investment outcomes are crucial indicators of performance for financial decision-making. In this exercise, each investment outcome is a point in the sample space, representing a different combination of success and failure between the two considered investments.
The exercise categorizes these outcomes through sets:\[O\] and \[M\]:
  • Set \(O\) includes outcomes where the oil investment is successful:
    \((S_s, M_s), (S_s, M_u)\).
  • Set \(M\) includes outcomes where the municipal bond investment is successful:
    \((S_s, M_s), (S_u, M_s)\).
The union \((O \cup M)\) contains any outcome where at least one investment succeeds:
\((S_s, M_s), (S_s, M_u), (S_u, M_s)\).
The intersection \((O \cap M)\) represents scenarios where both investments succeed:
\((S_s, M_s)\).
This systematic breakdown helps analysts assess risks, anticipate potential challenges, and balance their portfolios effectively. Understanding these outcomes improves strategic planning and future investment decisions.

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Most popular questions from this chapter

Assume that we have two events, \(A\) and \(B\), that are mutually exclusive. Assume further that we know \(P(A)=.30\) and \(P(B)=.40\) a. What is \(P(A \cap B) ?\) b. What is \(P(A | B) ?\) c. A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer. d. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities. \\[ \begin{aligned} P(\text { high-quality oil }) &=.50 \\ P(\text { medium-quality oil }) &=.20 \\ P(\text { no oil }) &=.30 \end{aligned} \\] a. What is the probability of finding oil? b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test follow. \\[ \begin{aligned} P(\text { soil } | \text { high-quality oil }) &=.20 \\ P(\text { soil } | \text { medium-quality oil }) &=.80 \\ P(\text { soil } | \text { no oil }) &=.20 \end{aligned} \\] How should the firm interpret the soil test? What are the revised probabilities, and what is the new probability of finding oil?

The prior probabilities for events \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(A_{1}\right)=.20, P\left(A_{2}\right)=.50,\) and \(P\left(A_{3}\right)=\) \(.30 .\) The conditional probabilities of event \(B\) given \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(B | A_{1}\right)=.50\) \(P\left(B | A_{2}\right)=.40,\) and \(P\left(B | A_{3}\right)=.30\) a. Compute \(P\left(B \cap A_{1}\right), P\left(B \cap A_{2}\right),\) and \(P\left(B \cap A_{3}\right)\) b. Apply Bayes' theorem, equation (4.19), to compute the posterior probability \(P\left(A_{2} | B\right)\). c. Use the tabular approach to applying Bayes' theorem to compute \(P\left(A_{1} | B\right), P\left(A_{2} | B\right)\) and \(P\left(A_{3} | B\right)\)

How many permutations of three items can be selected from a group of six? Use the letters \(A\), B, \(C, D, E,\) and \(F\) to identify the items, and list each of the permutations of items \(B, D,\) and \(F\)

An experiment with three outcomes has been repeated 50 times, and it was learned that \(E_{1}\) occurred 20 times, \(E_{2}\) occurred 13 times, and \(E_{3}\) occurred 17 times. Assign probabilities to the outcomes. What method did you use?
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