# Markov model to find the probability of a Pepsi drinker buying a Coke?

• Comp Sci
• shivajikobardan
In summary, a Markov model is a statistical model that can be used to determine the probability of a Pepsi drinker buying a Coke. This model takes into account the transition probabilities between different states, such as choosing one drink over the other. By analyzing past data and using the Markov model, the likelihood of a Pepsi drinker purchasing a Coke can be accurately predicted, providing valuable insights for businesses and marketing strategies.
shivajikobardan
Homework Statement
Find the probability of using coke for a current pepsi user in 4th purchase?Markov Model problem.
Relevant Equations
Transition Probability Matrix
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-:

This is the transition diagram.

This is the transition probability matrix-:

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-:

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-:

=[0.525 0.475]

Thus, it concludes that the required answer is 0.525.

Which one is correct in your opinion?

The two methods are identical, although the notation is apalling: who in their right mind uses superscript for indexation in this context? Do they use subscript for exponentiation?

The difference arises entirely from the interpretation of the question:
shivajikobardan said:
Find the probability of using coke for a current pepsi user in 4th purchase?
Does this mean ## p_0 = (0, 1) ## or ## p_1 = (0, 1) ## ? You interpret that as the former, and I agree with you, however the alternative solution has assumed the latter so what they are calling ## p^4 (!) ## is your ## p_3 ##.

FactChecker
pbuk said:
The two methods are identical, although the notation is apalling: who in their right mind uses superscript for indexation in this context? Do they use subscript for exponentiation?

The difference arises entirely from the interpretation of the question:

Does this mean ## p_0 = (0, 1) ## or ## p_1 = (0, 1) ## ? You interpret that as the former, and I agree with you, however the alternative solution has assumed the latter so what they are calling ## p^4 (!) ## is your ## p_3 ##.
The second TPM made by me was $TPM^4$
I am assuming that the rows are FROM and columns are TO
i.e row1, column1 is FROM Coke TO Coke i.e a coke eating guy again purchase coke.
But you're right. I've not seen a single textbooks about these examples so I'm also not sure why it's like that. I just tried to understand that pdf and solved accordingly.

I interpreted the question as you did, but as @pbuk points out, the book answer assumes that "a current pepsi user" implies that the first purchase was a pepsi. Therefore, it is asking for the probability after another 3 purchases. I think the book may be correct.
The important point is that your approach is correct. It is just a matter of being very careful of the problem statement.

## 1. What is a Markov model?

A Markov model is a mathematical model that is used to analyze the probability of a system transitioning from one state to another. It is based on the concept of a Markov chain, where the probability of transitioning to a certain state depends only on the current state and not on any previous states.

## 2. How does a Markov model apply to predicting a Pepsi drinker buying a Coke?

In this scenario, the Markov model would be used to analyze the probability of a person switching from being a Pepsi drinker to a Coke drinker. It would take into account various factors such as the person's age, location, and previous purchasing patterns to predict the likelihood of them switching to Coke.

## 3. What data is needed to create a Markov model for this situation?

To create a Markov model for predicting a Pepsi drinker buying a Coke, we would need data on the purchasing habits of both Pepsi and Coke drinkers, as well as demographic information such as age, location, and income. This data would be used to calculate the transition probabilities between the two states.

## 4. How accurate is a Markov model in predicting behavior?

The accuracy of a Markov model depends on the quality of the data used and the assumptions made in creating the model. In general, it can provide a good estimate of the probability of a person switching from being a Pepsi drinker to a Coke drinker, but it may not account for all possible factors that could influence the behavior.

## 5. Can a Markov model be used for other types of predictions?

Yes, Markov models can be applied to a wide range of scenarios where there is a sequence of events and the probability of transitioning from one event to another is of interest. They have been used in fields such as economics, finance, and biology to make predictions and analyze complex systems.

• General Math
Replies
1
Views
929
• Calculus and Beyond Homework Help
Replies
1
Views
4K
• General Math
Replies
2
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
11
Views
10K
• Special and General Relativity
Replies
1
Views
2K