Find the probability of using coke for a current pepsi user in 4th purchase?Markov Model problem.

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?

 

Your solution does not show,
but the probability transition matrix method is standard,
and given how the chain has small complexity (only two states)
I'd concur that is the way to go.

 

To determine the correct probability of a current Pepsi user purchasing Coke on the 4th purchase, let's clarify the Markov model and the transition matrix calculations step-by-step.

Markov Model and Transition Matrix

Given:

· Probability of purchasing Coke after Coke (C -> C): 80% or 0.8

· Probability of purchasing Pepsi after Coke (C -> P): 20% or 0.2

· Probability of purchasing Pepsi after Pepsi (P -> P): 70% or 0.7

· Probability of purchasing Coke after Pepsi (P -> C): 30% or 0.3

The transition probability matrix (TPM) is:

P = \begin{pmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \\ \end

Calculation Approach

Using Power of Transition Matrix

To find the state distribution after 4 purchases for a current Pepsi user, we can calculate P^4 and use the initial state vector representing a current Pepsi user.

Initial state vector:

p0= \begin{pmatrix} 0 & 1 \end

Let's computeP^4:

Compute P2=P.P

Compute P3=P^2.P

Compute P4=P3⋅P

However, the simplified solution method involves iteratively multiplying the initial state vector by the transition matrix.

Iterative Multiplication Approach

Given:

p0=\begin{pmatrix} 0 & 1 \end{pmatrix}

After 1st purchase:

P1=P0⋅P= \begin{pmatrix} 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \end{pmatrix}

= \begin{pmatrix} 0.3 & 0.7 \end{pmatrix}


After 2nd purchase:

P2=P1⋅P=\begin{pmatrix} 0.3 & 0.7 \end{pmatrix} \cdot \begin{pmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \end{pmatrix}

= \begin{pmatrix} 0.45 & 0.55 \end{pmatrix}

After 3rd purchase:

P3=P2⋅P=\begin{pmatrix} 0.45 & 0.55 \end{pmatrix} \cdot \begin{pmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \end{pmatrix}

= \begin{pmatrix} 0.525 & 0.475 \end{pmatrix}

After 4th purchase:

P4=P3⋅P=\begin{pmatrix} 0.525 & 0.475 \end{pmatrix} \cdot \begin{pmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \end{pmatrix}

= \begin{pmatrix} 0.5625 & 0.4375 \end{pmatrix}

Conclusion

Using the power of the transition matrix method, you correctly identified the value of 0.5625 from the second row and first column in P^4.

The iterative multiplication method confirms this value as well.

Thus, the correct probability of a current Pepsi user purchasing Coke on the 4th purchase is indeed 0.5625. The conflicting source seems to have a miscalculation in the iterative steps or in the matrix multiplication.


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