Modeling Real-World Probability Over Time|My homework helper

Posted: February 19th, 2023

 I need initial post and 2 responses to classmates.

Modeling Real-World Probability Over Time

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Suppose you and a team of coworkers are planning for the likelihood of two outcomes that may affect your company’s business in the future. You are tasked with modeling the interrelationship between two states, or outcomes, with a Markov system (i.e., state-transition diagram). Some possibilities would be the proportion of users who are paying for your service versus those who are not, devices that are produced to specification codes versus those that turn out defective, or kiosks that are working properly versus those that have started to malfunction.

Post 1: Initial Response

Develop a hypothetical scenario in which you and your team members are going to examine the likelihood that someone or something will be observed in one of two states at any point in time. As you develop your scenario, carefully address all of the following:

  1. In a brief narrative introducing your scenario, clearly identify two states for your scenario. Be sure to describe what the states are and define them as State A versus State B.
  2. Using the illustration provided as a guide, clearly present the probabilities associated with each of the four possible transitions, by labeling each of the transitions (indicated by the arrows) with an appropriate value for the probability they will occur (i.e., these are your own hypothetical estimates). Be sure these values adhere to probability norms regarding what range they may fall within and how they combine or relate with their paired probability. Share your complete state-transition diagram with a visual.
  3. Propose an initial distribution vector, v, which gives a hypothetical estimate of what proportion is initially observed in State A and State B. This should be reported as a 2 × 1 matrix and adhere to probability norms.

View Unit 10 Discussion Post 1 example.

Diagram consisting of two states and arrows indicating all the possible transitions to and from the states.

Post 2: Reply to a Classmate

Assume your team member has passed the diagram and initial setup on to you for further analysis of how the likelihood of these outcomes may affect the company in the short term. Review a classmate’s state-transition diagram and address all of the following items.

  1. Translate the probabilities given in the diagram to a transition matrix, P. Present this as a 2 × 2 matrix.
  2. Generate the proportion in State A versus State B after the first time period (i.e., the distribution after one step). This is also known as the state matrix after one period, S1, found by multiplying the initial distribution vector, v, given by your classmate by the transition matrix, P, you derived from their diagram (S1 = v∙P). Your result for S1 should be expressed as a matrix/vector with the appropriate dimensions.
  3. Interpret the state matrix after one period, S1, which you just computed, in your own words, using one or two sentences. In another one or two sentences, express to your team member some ideas on how the company might find this information useful.

View Unit 10 Discussion Post 2 example.

Post 3: Reply to Another Classmate

Assume your team members have now shared with you all of the information examined to date, including the diagram, initial setup, and analysis of what would be likely after one time period. Review a different classmate’s state-transition diagram and address all of the following items.

  1. Generate the proportions in State A versus State B for each period at least 20 periods from now, by selecting a value for n ≥ 20. Compute S2 and all subsequent distribution vectors for n steps into the future, by using the information presented by your classmates along with a computational tool of your choice (e.g., Excel®, Python®). Organize and present your results clearly as a summary table, showing the results for the distribution vectors for the first three periods and then also for the final three periods leading up to your nth state matrix/vector.
  2. What proportions are likely to be in State A and State B, long term? Address this question by interpreting the long-term trend you observe in the state matrices which you just computed, using one or two sentences. In another one or two sentences, express to your team members some ideas on how the company might find this information useful, relating it to your classmate’s original context.Matthew Goetzke

    1. Scenario:

    Big Tech Company is mapping the success rate of successful production of Unit A. Unit A has a 70% chance to pass quality and make it to the customer, and a 30% chance of being retained for repair. The customer has a 10% chance of sending Unit A back to Big Tech Company, and a 90% chance of retaining Unit A.

    2. Graphic

    Diagram, venn diagram  Description automatically generated

    3. Interpreted for one period

    State A = Unit A goes to quality

    State B = Unit A is at customer

    .3 * .3 +.1 * .7 = .16

    .9 * .7+.7 * .3 = .84

    V = [.16 , .84]

    This indicates that Units at quality have a 16% chance of being retained

     

     

    Jay Wilt posted Feb 17, 2023 10:52 AM

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    Hello Everyone,

    The application development center released a new product three months ago and the organization wants to see which users of the application are paying for accounts or how many accounts are vacant. Our records indicate that 20% of vacant accounts become paid user accounts while 80% stay vacant. The same records show another 10% of paid user accounts became vacant after a month while 90% stayed as paid user accounts.

    A: Paid user account

    B: Vacant account

    Diagram  Description automatically generated

    .90(.90) + .20(.10)= .83

    .80(.10) + .10(.90) = .17

    V =

    .83
    .17

     

    This means that an account has an 83% chance of staying as a paid user account during the initial vector versus the account becoming vacant.

SOLUTION

Scenario: Modeling the likelihood of an online shopper making a purchase

In this scenario, we will be examining the likelihood of an online shopper making a purchase, given the current state of their browsing behavior. We will define State A as the state in which the shopper is actively browsing products, and State B as the state in which the shopper has added items to their cart and is in the process of making a purchase.

State-transition diagram:

css
A B
+-------+ +-------+
A | | 0.2 | |
| +---->+ |
| | | |
+-------+ +-------+
0.8 0.8

The probabilities associated with each of the four possible transitions are as follows:

  • P(A to A) = 0.8: This represents the probability that a shopper will continue browsing products after viewing a product page, without adding any items to their cart.
  • P(A to B) = 0.2: This represents the probability that a shopper will add items to their cart and move to the purchase state, after viewing a product page.
  • P(B to A) = 0.8: This represents the probability that a shopper will abandon their cart and return to browsing products, after entering the purchase state.
  • P(B to B) = 0.2: This represents the probability that a shopper will complete their purchase and remain in the purchase state, after entering it.

Initial distribution vector: We can make a hypothetical estimate that initially, 70% of online shoppers are in the browsing state (State A), and 30% of them are in the purchasing state (State B). Thus, our initial distribution vector, v, can be represented as a 2×1 matrix:

css
v = [0.7, 0.3]'

Note that the sum of the elements in this vector is equal to 1, as it should be for any probability vector.

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