Designing a Markov Model for Coke and Pepsi Purchases

In summary, a Markov model and transition probability matrix were designed for the given data. Based on the model, the probability of a person purchasing coke after purchasing it once is 80%, and the probability of a person purchasing pepsi after purchasing it once is 70%. To find the probability of a pepsi user purchasing coke on their fourth purchase, two different approaches were taken. The first approach involved taking the TPM to the 4th power, resulting in a probability of 0.5625. The second approach involved using the formula for current distribution and taking it to the 4th power, resulting in a probability of 0.525. The correct answer is subjective and may depend on the interpretation of the data.
  • #1
shivajikobardan
674
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Design the markov model and transition matrix for the given data. Answer the following questions based on the mode.
a) If a person purchase coke now the probability of purchase of coke next time is 80%.
b) If a person purchases pepsi now the probability of purchasing pepsi next time is 70%.

Then,
Find the probability of using coke for a current pepsi user in 4th purchases-:

My solution-:
https://lh4.googleusercontent.com/4MgnCm3QHosELoWQk-S1rxJ1OjnuObR7yn80p1-UBH4TtJsvKVne-265fdK5QRGJmJ58hxTWn0zalcqpW3P43Bpew9jlbwF11IKw-HDr4aemdDutK2uLMben_diGd5Af3mY-bqgJ0Wen7o3eKGIJ9bA
This is the transition diagram.

This is the transition probability matrix-:

https://lh5.googleusercontent.com/26uw6G8iVZc4LPzBfp9Nq3aQq6j79F0QrJ8nPXHQrx2MRflA58zalCsQcxk8mnOWkUqWmP4j3Q_FNYGBtxwADr2eZsM0jUHTNZVpkHg4Y44SaMj888ccO-GRMvvQ0X-WotF14kKK8fa4T29CqNQ
So, what I did was basically to Took this TPM(Transition Probability Matrix) to the power 4. My basis for doing this was this source-: https://www.math.pku.edu.cn/teachers/xirb/Courses/biostatistics/Biostatistics2016/Lecture4.pdf

So what I got was-:

https://lh4.googleusercontent.com/3ez2gYlZOB1cji7ALC8QN5flVTZXih24-aGt1m4nIIpx2cM8hGoDvr3ZuD4AyJUQkJJIG7EBB-177CcorBfJXB9Qsniv92JvGHD0K2tDXdjLDJBMOrIea2wHw7iELOLlseZTVZ1_k0qxmkEdao2YGrE
Now I am assuming that the rows means FROM and column side means TO. And the first element of row and column is "Coke". So, to find from Pepsi to Coke, I'd go to second row and first column, the value would be 0.5625

But the problem is that, I've conflicting source which claims the answer is sth else-:

It solves it like this-:

P=TPM

p=Current distribution=[0 1]

Now, for 2nd purchase

p²=p*P=[0.3 0.7]

For 3rd purchase-:
p³=p² * P
=[0.45 0.55]

For 4th purchase-:
https://lh4.googleusercontent.com/VlMGXGT6dOGvTEHnelsVuW9Nr1NrPliK5WmhtPSzC86I9zOcj_Z70RzScMByMvnVdcS84flQAgfRLN1pFjpfisz62U06H4VZ_A9dwtswIXQQEelk8n02t87poHNOywHSbQ4dMAhJoggsd3cGMcs

=[0.525 0.475]

Thus, it concludes that the required answer is 0.525.

Which one is correct in your opinion?
 
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FAQ: Designing a Markov Model for Coke and Pepsi Purchases

1. What is a Markov model?

A Markov model is a mathematical model that uses probability to predict the future state of a system based on its current state. It is used to model systems that involve random processes and can be used to understand and predict consumer behavior.

2. Why is a Markov model useful for studying Coke and Pepsi purchases?

A Markov model is useful for studying Coke and Pepsi purchases because it allows us to analyze the probability of a consumer switching between these two brands based on their previous purchases. This can help companies understand consumer preferences and make informed marketing and advertising decisions.

3. How is a Markov model designed for Coke and Pepsi purchases?

A Markov model for Coke and Pepsi purchases is designed by first identifying the different states of the system, such as purchasing Coke, purchasing Pepsi, or not purchasing either. Then, transition probabilities are calculated based on historical data to determine the likelihood of a consumer switching between these states. The model can be refined and adjusted as needed to accurately reflect consumer behavior.

4. What data is needed to create a Markov model for Coke and Pepsi purchases?

To create a Markov model for Coke and Pepsi purchases, data on consumer purchase behavior is needed. This includes information on the number of purchases, the brands purchased, and the sequence of purchases. Additional data, such as demographic information and advertising spending, can also be incorporated to improve the accuracy of the model.

5. What are some limitations of using a Markov model for Coke and Pepsi purchases?

One limitation of using a Markov model for Coke and Pepsi purchases is that it assumes that consumer behavior is solely based on their previous purchases and does not take into account other factors, such as brand loyalty or external influences. Additionally, the model may not accurately reflect sudden changes in consumer behavior or new market trends. It is important to regularly update and validate the model to ensure its effectiveness.

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