Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

List the simple events associated with each experiment. Data concerning durable goods orders are obtained each month by an economist. A record is kept for a 1 -yr period of any increase \((i)\), decrease \((d)\), or unchanged movement \((u)\) in the number of durable goods orders for each month as compared with the number of such orders in the same month of the previous year.

Step-by-Step Solution

Answer

The simple events for this experiment are all possible combinations of increases (i), decreases (d), and unchanged movements (u) in the monthly durable goods orders over a one-year period (12 months). There are 3 possible outcomes for each month, resulting in a total of \( 3^{12} \) simple events for this experiment.

Step 1: Understanding the experiment

In this problem, we have an experiment where monthly durable goods orders are observed for a one-year period. For each month, there are three possible outcomes: the number of orders increases (i), decreases (d), or remains the same (u) compared to the same month's orders the previous year.

Step 2: Identifying total months

In a year, there are 12 months. We need to find the possible simple events for each month, considering the three possible outcomes for each month's record.

Step 3: Listing simple events for each month

As we have 3 outcomes (i, d, and u) for each month and 12 months in a year, we can consider every combination of these outcomes for each month. We can list all the simple events for a one-year period as a sequence \( (E_1, E_2, E_3, ..., E_{12}) \), where each \( E_i \) can take one of the possible outcomes, i.e., i, d, or u. Based on this idea, there are \( 3^{12} \) possible combinations of outcomes for the 12 months.

Step 4: Summary of the results

In conclusion, we have identified that there are 3 possible outcomes (i, d, and u) for each of the 12 months in a year, which gives a total of \( 3^{12} \) simple events for this experiment with durable goods orders.

Key Concepts

Simple Events

In probability, a simple event is an outcome or a combination of outcomes that cannot be decomposed further. When examining monthly durable goods orders, each month's result—be it an increase, decrease, or unchanged count—is considered a simple event.
This is because each of these outcomes represents the most fundamental level of detail in this context.
Simple events are vital because they form the building blocks of more complex events in probability.
By understanding the most basic level of outcomes, we can compute the probabilities of larger events that might include multiple months or a series of fluctuations in durable goods orders. This simplification makes calculating probabilities more manageable. To summarize, in our exercise with durable goods orders:

• Each month's outcome is a simple event.
• The outcomes are: increase (i), decrease (d), and unchanged (u).
• Simple events help us understand more complex phenomena by breaking them down into basic parts.

Outcomes

Outcomes are the possible results of an experiment. In any statistical experiment, identifying the range of possible outcomes is crucial because it allows us to calculate probabilities and predict future events. In our example of durable goods orders, understanding outcomes is directly connected to recognizing the impact of economic trends over the year.
For each month, the outcomes can be:

• Increase in orders compared to the previous year (i)
• Decrease in orders (d)
• No change in order levels (u)

Given that there are 12 months in a year and 3 possible outcomes for each month, calculating outcomes involves acknowledging each individual month's situation separately and together as a pattern over time.
Therefore, the full set of outcomes for one year consists of all possible permutations of these three states for every month, amounting to different potential sequences of monthly outcomes. This comprehensive perspective helps economists understand trends and fluctuations in durable goods orders over time.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arranging, and finding patterns, especially essential when dealing with various outcomes. It allows us to calculate the number of possible outcomes and sequences that can arise from a particular setup.
In the context of our durable goods orders scenario, combinatorics enables us to figure out the number of different possible sequences of outcomes over the 12 months.
Given that each month can have 3 different outcomes (increase, decrease, unchanged), combinatorics reveals the total number of combinations by 3 choices each month across a year.We use the formula for the total number of possible sequences:

• The total combinations are calculated as \( 3^{12} \).
• This represents all different sets of outcomes over the period of a year.
• Combinatorics simplifies dealing with large numbers of possibilities.

It's particularly helpful in understanding the vast array of potential economic situations that could occur, and helps in forming predictions and planning based on different hypothetical scenarios.