1. models that's involves chance or risk, often measured as probability value are_______.

2. in many cases ______ ____ _____ is the most important and the most difficult step in quantitative anaysis approach.
3. ______ should be solvable, realistic, and easy to understand and modify and the required input data should be obtainable.
4. what are the four potential roadblocks that quantitative analysis face in defining a ______.
5. the four criteria for decision making under certainty can be computed directly from the _____ ____.
Please help

1. models that involve chance or risk, often measured as probability value are called probabilistic models. These models are used to analyze situations where outcomes are uncertain and involve probabilities.

To find examples of probabilistic models, you can start by looking into fields such as finance, insurance, and operations research. In finance, for example, options pricing models like the Black-Scholes model involve calculating probabilities of different future stock price movements. In insurance, actuarial models are used to assess and predict risks based on historical data and probability distributions. In operations research, simulation models are used to simulate uncertain events and their impact on system performance.

2. In many cases, problem formulation is the most important and the most difficult step in the quantitative analysis approach. Problem formulation involves defining the objective of the analysis, identifying the variables and parameters involved, and stating any constraints or limitations.

To effectively address problem formulation, you can follow these steps:
a) Clearly define the problem and its objectives. What are you trying to achieve with your analysis?
b) Identify all the necessary variables and parameters. What are the factors that influence the problem?
c) Consider any constraints or limitations. Are there any restrictions or conditions that need to be taken into account?
d) Decide on the appropriate level of detail. How granular should the analysis be?
e) Seek input and feedback from relevant stakeholders. Are there any perspectives or insights that should be considered?

3. A quantitative model should be solvable, realistic, and easy to understand and modify, and the required input data should be obtainable. Ensuring these qualities is important for the model's effectiveness and usability.

To ensure solvability, you need to select appropriate mathematical methods and techniques that are capable of finding solutions to the problem at hand. This may involve techniques such as linear programming, optimization algorithms, or statistical methods.

To make the model realistic, you should include all relevant variables and parameters that accurately represent the problem being analyzed. Consider the assumptions and simplifications you make, and make sure they align with the real-world context.

Easy understandability and modifiability of the model mean that it should be clear and straightforward to interpret and modify. Use documentation, proper labeling, and clear annotations to make the model understandable by others. Ensure that it can be easily adapted and updated as new data or changes in the problem occur.

Obtaining the required input data means having access to accurate and reliable data that is necessary to run the model. In some cases, this may involve collecting historical data or conducting surveys and experiments. It is important to consider data quality and availability when designing a quantitative model.

4. Four potential roadblocks that quantitative analysis may face in defining a problem are:

a) Lack of data: Insufficient or low-quality data can hinder the accuracy and reliability of the analysis. This may require additional data collection efforts or adjusting the scope of the analysis.

b) Complex or dynamic systems: Some problems involve complex or dynamic systems that are difficult to model accurately. This could be due to factors such as nonlinear relationships, feedback loops, or multiple interacting variables.

c) Uncertainty and risk: If there is a high level of uncertainty or risk associated with the problem, it can be challenging to quantify and incorporate these factors into the analysis. Probabilistic modeling techniques may be required to address this issue.

d) Stakeholder preferences and biases: The preferences and biases of stakeholders involved in the analysis can affect how the problem is defined and approached. It is important to consider these factors and ensure a balanced and objective analysis.

5. The four criteria for decision making under certainty can be computed directly from the decision matrix. The decision matrix is a tool used to evaluate alternatives based on multiple decision criteria.

The criteria for decision making under certainty are:

a) Maximax: This criterion seeks to maximize the maximum payoff. To compute this criterion, identify the maximum payoff for each alternative and select the alternative with the highest maximum payoff.

b) Maximin: This criterion seeks to maximize the minimum payoff. To compute this criterion, identify the minimum payoff for each alternative and select the alternative with the highest minimum payoff.

c) Minimax regret: This criterion seeks to minimize the maximum regret. Regret is the difference between the best payoff for a given alternative and the actual payoff. To compute this criterion, calculate the regret for each alternative and select the alternative with the lowest maximum regret.

d) Equal likelihood (Laplace): This criterion assigns equal probabilities to all possible outcomes and seeks to maximize the average payoff. To compute this criterion, calculate the average payoff for each alternative and select the alternative with the highest average payoff.

These criteria can be computed directly from the decision matrix by considering the payoffs for each alternative and the probabilities associated with each outcome.