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What is the Application of Matrices in Markov Chains (Steady State)?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
Matrices help us find the 'steady state' in Markov Chains. The steady state is a special condition where the probabilities of being in different states no longer change over time, even if the system keeps running. Matrices allow us to calculate these stable, long-term probabilities.
Simple Example
Quick Example
Imagine a mobile network where users switch between 'Active' and 'Inactive' states. If 80% of active users stay active and 20% go inactive, and 30% of inactive users become active while 70% stay inactive, we can use matrices to find what percentage of users will eventually be active and inactive in the long run, no matter where they start.
Worked Example
Step-by-Step
Let's say a chai shop has two types of customers: 'Regular' (R) and 'New' (N). Each day:
- 70% of Regulars stay Regular, 30% become New.
- 40% of New customers become Regular, 60% stay New.
We want to find the steady state probabilities.
Step 1: Write the transition matrix (P).
P = [[0.7, 0.3],
[0.4, 0.6]] (Rows are 'from', columns are 'to')
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Step 2: Let the steady state probability vector be S = [p1, p2], where p1 is the probability of being Regular and p2 is the probability of being New. In steady state, S * P = S.
So, [p1, p2] * [[0.7, 0.3],
[0.4, 0.6]] = [p1, p2]
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Step 3: This gives two equations:
0.7*p1 + 0.4*p2 = p1
0.3*p1 + 0.6*p2 = p2
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Step 4: Simplify the first equation:
0.4*p2 = 0.3*p1 => 4*p2 = 3*p1 => p1 = (4/3)*p2
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Step 5: We also know that the sum of probabilities must be 1: p1 + p2 = 1.
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Step 6: Substitute p1 from Step 4 into the equation from Step 5:
(4/3)*p2 + p2 = 1
(4/3 + 1)*p2 = 1
(7/3)*p2 = 1
p2 = 3/7
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Step 7: Find p1 using p1 = (4/3)*p2:
p1 = (4/3) * (3/7) = 4/7
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Answer: In the steady state, 4/7 (approx 57.14%) of customers will be Regular, and 3/7 (approx 42.86%) will be New.
Why It Matters
Understanding steady states helps predict long-term behavior in many fields. In AI/ML, it's used in recommendation systems; in FinTech, for stock market predictions; and in Biotechnology, for modeling disease spread. This knowledge can lead to careers in data science, financial analysis, or medical research.
Common Mistakes
MISTAKE: Multiplying the probability vector by the transition matrix in the wrong order (P * S instead of S * P). | CORRECTION: The probability vector is usually a row vector, so it pre-multiplies the transition matrix (row vector * matrix).
MISTAKE: Forgetting that the sum of steady state probabilities must always be 1. | CORRECTION: Always include the equation 'p1 + p2 + ... + pn = 1' when solving for steady state probabilities, as it's crucial for finding a unique solution.
MISTAKE: Not correctly setting up the equations from S * P = S. | CORRECTION: Each element of the resulting vector (S) should be equal to the corresponding element of the original vector (S), leading to a system of linear equations.
Practice Questions
Try It Yourself
QUESTION: A town has two weather states: Sunny (S) and Cloudy (C). If it's Sunny today, there's a 70% chance it's Sunny tomorrow and 30% chance it's Cloudy. If it's Cloudy today, there's a 60% chance it's Sunny tomorrow and 40% chance it's Cloudy. Find the steady state probability of a Sunny day. | ANSWER: 2/3 (approx 66.67%)
QUESTION: A student can be in one of two moods: Happy (H) or Sad (S). If Happy, they stay Happy with 80% chance, become Sad with 20%. If Sad, they become Happy with 40% chance, stay Sad with 60%. What are the steady state probabilities for being Happy and Sad? | ANSWER: Happy = 2/3, Sad = 1/3
QUESTION: A company's product has two market states: 'Popular' (P) or 'Niche' (N). Each year, 90% of Popular products stay Popular, 10% become Niche. 50% of Niche products become Popular, 50% stay Niche. What is the long-term percentage of products that will be Popular? | ANSWER: 5/6 (approx 83.33%)
MCQ
Quick Quiz
Which property is true for a system in a steady state of a Markov Chain?
The probabilities of being in each state continuously change.
The probabilities of being in each state remain constant over time.
The system always returns to its initial state.
The transition matrix changes with each step.
The Correct Answer Is:
B
In a steady state, the probabilities of being in different states do not change over time, meaning the system has reached a stable distribution. Options A, C, and D describe scenarios that are not characteristic of a steady state.
Real World Connection
In the Real World
In India, companies like Flipkart or Amazon use Markov Chains to model customer purchasing behavior. They can predict, using steady state analysis, what percentage of customers will eventually be 'frequent buyers' versus 'occasional buyers' based on their past actions. This helps them plan marketing strategies and inventory, making your online shopping experience smoother!
Key Vocabulary
Key Terms
Markov Chain: A mathematical model describing a sequence of events where the probability of each event depends only on the state of the previous event. | Steady State: A condition in a Markov Chain where the probabilities of being in different states no longer change over time. | Transition Matrix: A square matrix describing the probabilities of moving from one state to another in a Markov Chain. | Probability Vector: A row vector whose elements are the probabilities of being in each state, and sum up to 1.
What's Next
What to Learn Next
Great job learning about steady states! Next, explore how Markov Chains are used in more complex scenarios, like page ranking algorithms (how Google decides which websites show up first!). This will show you even more powerful applications of these concepts.
