Stationary distribution of a Markov chain refers to a probability distribution where the probabilities of being in each state remain constant over time. It represents the long-term behavior of the chain and is reached when the initial conditions no longer affect the distribution. The stationary distribution is determined by the transition probabilities and the initial distribution. It is crucial for understanding the asymptotic behavior of the chain and is used in various applications, including modeling queueing systems, analyzing population dynamics, and designing simulation algorithms.
Define Markov chains and explain their basic principles.
Markov Chains: A Tale of Unpredictable Journeys
Picture yourself lost in a labyrinth of uncertain possibilities, where every step you take is influenced by an invisible hand. Welcome to the fascinating world of Markov chains, where the past holds the key to the future.
Unveiling Markov’s Magic
Imagine rolling a dice repeatedly, where the outcome of each roll determines the next roll. Markov chains capture this idea by modeling sequences of events where the probability of the next event depends solely on the current event, not the entire history. It’s like a magic hat that spits out probabilities based on where you are right now.
State Space: The Playground of Possibilities
The journey begins in a state space, a collection of all possible states the chain can occupy. Each state represents a different scenario, like sunny or rainy weather, or the mood of your grumpy cat.
Transition Matrix: The Probability Map
A transition matrix is the secret recipe that governs the chain’s behavior. It holds the probabilities of moving from one state to another. Think of it as a roadmap that shows you the chances of jumping from “rainy” to “sunny” or from “catnip high” to “indifferent.”
Initial Conditions: Setting the Stage
Every adventure has a starting point. In Markov chains, initial conditions determine the starting state. Imagine choosing a random door to enter a haunted house, not knowing what lurks behind it. That’s the power of initial conditions.
Properties of Markov Chains: Unraveling the Patterns
Markov chains come with a bag of tricks, each defining their unique characteristics.
- Ergodicity: Can the chain roam freely between all states? Think of a restless wanderer who keeps exploring new lands.
- Recurrence: Will the chain revisit certain states over and over? It’s like an emotional rollercoaster with ups and downs.
- Asymptotic stability: Does the chain eventually settle into a steady state? Imagine reaching a comfortable equilibrium like a peaceful pond after a storm.
- Irreducibility: Can the chain move between any two states without getting stuck? It’s like having an all-access pass to explore every corner of the labyrinth.
- Aperiodicity: Do states have a predictable pattern of visitation? Think of a Ferris wheel that always stops at the same spot.
- Positive recurrence: Do states have a guaranteed return time? It’s like a boomerang that always finds its way back to you.
Markov Property and Chapman-Kolmogorov Equations: Unlocking the Secrets
Markov property is the key to understanding Markov chains: the future is shaped by the present, not the past. It’s like a superhero that predicts the future by only looking at the here and now.
Chapman-Kolmogorov equations are the mathematical tools that allow you to calculate the probabilities of the chain’s journey. Think of them as a GPS that guides you through the labyrinth of states.
Describe the concept of state space and transition matrix.
Markov Chains: Unveiling the Secrets of Random Walks
State Space: Where the Action Unfolds
Imagine a coin flip – two possible outcomes: heads or tails. Now, let’s flip the coin repeatedly. The sequence of heads and tails represents the state space of a Markov chain, the set of all possible outcomes that your coin can land on.
Transition Matrix: Mapping the Journey
With each flip, the coin transitions from one state to another. These transitions are captured by the transition matrix. It’s like a roadmap, telling us the probability of transitioning from one state (heads or tails) to another. For our coin, the transition matrix is a 2×2 grid:
| Heads | Tails |
|---|---|
| 0.5 | 0.5 |
| 0.5 | 0.5 |
This matrix shows that the probability of transitioning from heads to heads or tails is 0.5, and the same for tails.
Ergodicity: Will it Wander or Get Stuck?
Will your coin visit all states infinitely often? That’s where ergodicity comes in. An ergodic chain is like a footloose wanderer, eventually visiting every outcome. Our coin flip is a prime example of an ergodic chain – it can’t get stuck on one side!
Recurrence: The Relentless Return
Recurrence is another quirky property: will a state keep popping up over and over? Our coin flip is positive recurrent, meaning that it will visit both heads and tails infinitely often.
Irreducibility: The Chain’s Unstoppable Flow
Irreducible chains are the free spirits of the Markov world. They can move between any two states without getting trapped. Our coin flip is once again the star of the show, with its ability to switch between heads and tails with ease.
Aperiodicity: Breaking the Cycle
Finally, aperiodicity means that a state doesn’t have a regular pattern of visitation. Our coin flip is unpredictable in this regard, since it doesn’t favor one side over the other.
So, there you have it – a crash course in Markov Chains, the whimsical world of randomness and probability. From state spaces to transition matrices and quirky properties, these chains bring a touch of mathematical magic to the unpredictable realm of random events.
Demystifying Markov Chains: A Journey Through Probabilistic Timelines
Markov chains are like a game of probability where the next move depends only on your current location. They’re like a dice with memory, where the number you roll today is influenced by the ones you’ve rolled before.
2. State Space and Transition Matrix
Think of the game board as the state space, and the dice rolls as transitions. Each state represents a possible outcome, and the transition matrix tells you the likelihood of moving from one state to another. It’s like a treasure map that guides your probabilistic journey.
3. Discuss Transition Probabilities and Their Role in Defining the Behavior of the Chain
Now, let’s talk about the real magic: transition probabilities! These are the weights on your probability dice. They determine how often you’re likely to land on each state. High transition probabilities mean you’ll visit that state more frequently, while low ones mean it’s a less common destination.
The pattern of transition probabilities shapes the behavior of your Markov chain. Like a secret code, they dictate where you’re headed next, leading you through a unique path of probabilistic adventure.
4. Initial Conditions
Before you start rolling the dice, you need to set the starting point. That’s where initial conditions come in. They tell you the probability of being in each state initially. It’s like choosing the first card from your deck, determining the hand you’re dealt.
5. Properties of Markov Chains
Markov chains have some pretty cool properties. Here are a few key ones:
- Ergodicity: Will you eventually wander into all corners of the state space?
- Recurrence: Are there states you’re destined to revisit endlessly?
- Asymptotic stability: Will you eventually settle into a steady state, like a ship finding its equilibrium?
- Irreducibility: Can you hop between any two states like a hopping kangaroo?
- Aperiodicity: Do you have a repeating pattern of state visits, like a dancing pendulum?
- Positive recurrence: Will you inevitably return to certain states, like a boomerang that always finds its way back?
6. Markov Property and Chapman-Kolmogorov Equations
The Markov property is the secret sauce that makes Markov chains so predictable. It says that your future is only concerned with your present state, like a fortune teller who only needs to know your birthday.
The Chapman-Kolmogorov equations are like mathematical superheroes that calculate the transition probabilities for any sequence of states. They’re the secret formula that unlocks the mysteries of Markov chains.
So, there you have it! Markov chains: a probabilistic rollercoaster that reveals the hidden patterns in time and space. From games of chance to weather forecasting, they’re a powerful tool for exploring the unpredictable with a touch of mathematical magic.
Understanding Markov Chains: A Beginner’s Guide
Hey there, folks! Welcome to the fascinating world of Markov chains, where the future is shaped by the present and past. In this blog post, we’ll unravel the secrets of these mathematical gizmos that help us understand real-life phenomena, from weather patterns to internet browsing habits.
What’s a Markov Chain?
Picture this: you’re playing a game of hopscotch. Each square you land on determines where you go next, but your previous hops don’t matter. That’s the basic idea behind a Markov chain: it’s a sequence of events where the probability of the next event depends only on the current event.
Initial Conditions: Setting the Stage
Initial conditions are like the starting point of aMarkov chain. They tell us what state the chain is in at the beginning. It’s crucial because it sets the stage for the chain’s future evolution. The initial probability distribution tells us the probability of being in each state initially, and the stationary distribution shows us the long-term, steady-state behavior of the chain.
Key Properties: Unraveling the Dynamics
Markov chains have some mind-boggling properties that govern their behavior. Let’s dive into the most important ones:
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Ergodicity: This property tells us whether the chain will eventually visit all possible states, or if it’s doomed to stay stuck in a few.
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Recurrence: It shows us if the chain will ever return to certain states infinitely often, or if they’ll become a distant memory.
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Asymptotic stability: This one lets us know if the chain will eventually settle down into a steady-state distribution, or if it’ll keep hopping around forever.
Markov Property: The Past Doesn’t Matter
Here’s the real magic of Markov chains: the future of the chain is independent of its past, given the present state. It’s like you’re starting fresh with every step you take. This property is so powerful that it’s used to model everything from weather forecasts to financial time series.
Chapman-Kolmogorov Equations: The Rules of Transition
These equations are like the secret formula that governs how Markov chains move between states. They tell us how to calculate the probability of transitioning from one state to another, taking into account all the possible paths in between.
Markov chains are a fascinating tool for understanding the dynamics of real-world phenomena. They’re used in a wide range of applications, from speech recognition to gene sequencing. So, next time you’re wondering why the weather keeps surprising you or why your favorite website seems to show you the same ads over and over again, remember the power of Markov chains!
Define initial probability distribution and stationary distribution.
Markov Chains: The Memory Makers for Data
Picture this: you’re at a party, and everyone’s wearing a different color shirt. You notice that the person you’re talking to is wearing a red shirt. What’s the probability that the next person you talk to will also be wearing red?
Enter Markov Chains!
These magical mathematical models are like party detectives, keeping track of the color of your previous shirt conversation to predict the color of your next one. They’re like “memory keepers” for data, helping us understand how things change over time.
State Your Space
Think of the different shirt colors as “states” in our Markov chain. And just like how you can’t time-travel to last week’s party, Markov chains only care about the current state, not the past ones.
Transition Matrix: The Party Dance Floor
This matrix is like a dance floor where states can transition from one to another. It shows the probability of moving from one color to another. So, if 30% of people at the party are wearing red shirts, your transition matrix would show a 30% chance of talking to another red-shirted person.
Initial Probability: The First Dance
When you walk into the party, you don’t know who you’ll talk to first. That’s where the initial probability distribution comes in. It tells us the likelihood of starting with a particular state (color).
Stationary Distribution: Drumroll The Grand Finale
As the party goes on, you might notice that the proportion of people wearing different colors stabilizes. This is called the stationary distribution. It shows us the long-term behavior of the chain, revealing the most popular shirt color of the night!
So, there you have it – Markov chains, the secret agents of data that help us understand the past, predict the present, and prepare for the future. Now get out there and start predicting shirt colors!
Markov Chains: Delving into the World of Mathematical Wonder
Hello there, curious minds! Today, we’re taking a whimsical journey into the fascinating world of Markov chains. These mathematical marvels can predict the future based solely on the present, like a magical time machine that knows exactly where you’ll be next. Let’s dive right in!
Key Properties of Markov Chains
Markov chains possess an enchanting array of properties that make them incredibly versatile tools:
Ergodicity: Imagine a chain hopping from state to state. If it eventually visits every state infinitely often, we say it’s ergodic. It’s like a restless explorer who never misses a single corner of the world.
Recurrence: Some states in a chain are like comfortable armchairs you just can’t resist. If you start from one of these recurrent states, you’ll keep coming back over and over again. Think of it as a cozy reunion with an old friend.
Asymptotic Stability: Markov chains can sometimes reach a steady state, known as asymptotic stability. It’s like a pendulum that eventually settles down to a rhythmic swing. The probability of being in a particular state becomes constant as time goes on.
Irreducibility: If a chain can leap from any state to any other, it’s said to be irreducible. It’s like a skilled dancer who can effortlessly transition between different moves.
Aperiodicity: Some states have a regular pattern of visitation, like a clock ticking away. This is known as aperiodicity. But if the visits are more sporadic, the chain is considered periodic.
Positive Recurrence: States in a chain can have a finite expected time to visit. It’s like a favorite café you keep returning to at regular intervals. This property is known as positive recurrence.
Understanding Markov Chains: A Journey into Chance Encounters
Imagine a mischievous elf named Markov who hops around a magical kingdom, governed by a whimsical set of rules. These rules define the realm of Markov chains, where the future is shaped solely by the present.
Ergodicity: Does Our Elf Dance Through the Whole Kingdom?
Markov is an adventurous spirit, eager to explore every nook and cranny of the kingdom. Ergodicity measures his ability to eventually visit all states. If a chain is ergodic, like Markov, it’s a true nomad, wandering freely and leaving no stone unturned.
Consider a kingdom with two castles, Castle A and Castle B. Markov starts in Castle A. If he can move between the castles without restrictions, the chain is ergodic. Each castle has an equal chance of being visited, regardless of where Markov starts.
But what if there’s a moat around Castle B, preventing Markov from crossing over? In this case, the chain is not ergodic. Markov would be stuck in the confines of Castle A, unable to experience the delights of Castle B.
Recurrence: Will Our Elf Ever Return?
Once Markov visits a state, he may choose to return later. Recurrence measures how often a state is visited again and again. If a state is recurrent, it means Markov will inevitably find his way back.
In our castle example, let’s say Castle A has a magnetic draw for Markov. Every time he ventures out to Castle B, he’s compelled to return to Castle A. This state (Castle A) is recurrent, constantly drawing Markov back.
Asymptotic Stability: Does Our Elf Settle Down?
As Markov dances through the kingdom, he may eventually settle down in one particular state. Asymptotic stability measures whether the chain converges to a steady-state distribution, where certain states become more likely to be visited.
Imagine a kingdom with three castles: A, B, and C. Markov starts in Castle A. After exploring, he finds that he spends most of his time in Castle B. As the chain progresses, Castle B’s probability of being visited approaches 100%. This means the chain is asymptotically stable, with Markov residing primarily in Castle B.
By understanding these properties, you can predict the behavior of Markov chains, unraveling the mysteries of chance encounters and guiding your journey through the realm of probability.
Markov Chains: Time to Start Recurring!
Picture this: you’re playing a game of “States and Chances” with a coin. Each flip of the coin determines your next state—either heads or tails. Now, here’s the twist: your future flips depend solely on the current one. This, my friends, is the essence of a Markov chain.
Imagine every coin flip as a state in a chain of events. Each state represents a possible outcome. But hold on tight, because the transition matrix is the real star of the show. It tells you how likely you are to jump from one state to another.
In our coin-flipping game, the transition matrix would look like this:
| Current State | Next State (Heads) | Next State (Tails) |
|---|---|---|
| Heads | 0.6 | 0.4 |
| Tails | 0.3 | 0.7 |
This means the coin has a 60% chance of landing on heads after a heads flip and a 40% chance of landing on tails. Similarly, after a tails flip, the odds are 30% heads and 70% tails.
Now, let’s talk about recurrence. It’s about the chain’s love affair with certain states. Recurrence asks the burning question: does the chain keep coming back to specific states over and over, infinitely often?
In our coin-flipping game, the chain will recur to both heads and tails. No matter how many flips you make, you’ll eventually land on either heads or tails. In the long run, you’ll dance between these two states, like a yo-yo!
So, recurrence is like a moth drawn to a flame—the chain can’t resist the allure of certain states. And that’s the magical mystery of Markov chains, my friends!
Asymptotic stability: Does the chain approach a steady-state distribution?
Markov Chains: Your Guide to Understanding the Future
Imagine you’re flipping a coin. Each toss is independent of the last, right? Not so with Markov chains, my friends! These bad boys have a memory.
What’s a Markov Chain?
It’s like a naughty chain that only remembers its last link. The current state of the chain influences the next, just like your grumpy boss’s mood can influence how you feel at work.
State Space and Transition Matrix
The states are like the different moods of your boss (or your cat). The transition matrix tells you how likely you are to go from one state to another. So, if your boss is currently happy, the matrix might say there’s a 90% chance they’ll still be happy tomorrow. But if they’re grumpy, oof, only a 10% chance for sunshine.
Initial Conditions
It all starts with a little push. The initial probability distribution tells the chain which state it’s starting in. Like, if you start your day with a coffee, you’re probably setting the tone for the rest of it.
Asymptotic Stability: Settling Down
Over time, some Markov chains settle into a steady-state distribution. It’s like your boss eventually getting used to your terrible jokes and becoming more tolerable. The chain might still bounce around a bit, but it’ll tend to hang out in this distribution.
Other Cool Properties
Markov chains have all sorts of other quirks. Like ergodicity (does it visit all states?), recurrence (does it keep coming back to certain states?), and irreducibility (can it move between any two states?). These properties tell us more about the chain’s behavior.
Markov Property and Chapman-Kolmogorov Equations
The Markov property is like a chain’s motto: “Forget the past, only the present matters!” It means the future of the chain depends only on its current state. And the Chapman-Kolmogorov equations? They’re like fancy formulas that calculate the probabilities of moving between states based on the Markov property.
Markov chains are awesome! They’re like little stories that unfold before our eyes, with each step influenced by what came before. From weather patterns to stock prices, they’re used in all sorts of fields to predict the unpredictable. So, if you want to peek into the future, grab a Markov chain and let it take you on an adventure!
Irreducibility: The Ability to Dance Between States
Imagine a Markov chain as a party where guests can mingle freely. Irreducibility means that every guest can dance with every other guest, no matter how introverted or extroverted they are. It’s like a well-connected social network where everyone can chat up anyone, anytime.
Technically speaking, irreducibility means that there are no isolated groups or cliques within the chain. Every state has a path to reach every other state, like a giant web of interconnected possibilities. This property is crucial because it ensures that the chain can explore its entire state space without getting stuck in a rut.
In real-world applications, irreducibility can be a vital factor. For instance, in a Markov chain modeling website navigation, irreducibility guarantees that users can freely navigate between any page without hitting dead ends. In a Markov chain forecasting weather patterns, irreducibility ensures that the model can capture the unpredictable nature of weather and transition smoothly between different states, like from sunny to rainy.
Aperiodicity: Do states have a regular pattern of visitation?
Aperiodic Markov Chains: When States Dance to Their Own Beat
Imagine a whimsical state machine, like a Markov chain, where states waltz around like carefree dancers. What if, instead of following a predictable pattern like a square dance, these states had a quirky habit of hopping and skipping in an unpredictable manner? That’s the essence of aperiodic Markov chains.
Now, let’s get a little technical. Aperiodic Markov chains are characterized by the states’ lack of periodicity, which means they don’t settle into a rhythmic pattern of visitation. It’s like watching a jazz improvisation session where the melody meanders through different chords, never repeating the same sequence twice.
Unlike ballroom dancers, who stick to a rigid waltz beat, aperiodic Markov chains are free spirits. They defy the notion of a predictable loop, introducing an element of randomness into their journey. Think of it as a spontaneous dance party where even the most seasoned dancer can’t predict the next move.
Aperiodic vs. Periodic Markov Chains
To understand aperiodicity, let’s contrast it with periodic Markov chains. Periodic chains have states that cycle through a regular pattern of visitation. Picture a group of states performing a synchronized dance, each taking turns to step forward. In periodic chains, the least common multiple of the periods of all states determines the overall pattern.
Aperiodic Markov chains, on the other hand, lack this synchronized rhythm. Their states dance to their own tune, sometimes revisiting each other quickly, other times taking extended breaks before returning. It’s like a dance floor where individuality reigns supreme.
The Significance of Aperiodicity
Aperiodicity is not just a quirky characteristic; it has practical implications in various fields. For instance, in queuing theory, it can help predict the behavior of systems where events occur randomly, such as arrivals and departures at a bus stop. In reliability engineering, aperiodicity can provide insights into the long-term behavior of complex systems, such as the maintenance schedules of equipment.
So, the next time you encounter a Markov chain, don’t expect a predictable waltz. Embrace the aperiodic dance of its states, where randomness reigns and patterns are left to the imagination. After all, even a chaotic dance can be captivating if you let yourself go and enjoy the unpredictable journey.
Positive recurrence: Do states have a finite expected time to visit?
Introducing Markov Chains: Unraveling the Path of Probability
Picture this: you’re flipping a coin, wondering where each toss will lead you – heads or tails? Well, Markov chains are like that, but on a much grander scale. They’re a way to predict how a system evolves over time, based on its current state and a dash of probability. Let’s dive in!
State Space and Transition Matrix: The Map of Possibilities
Imagine a row of boxes, each representing a different state your system can be in. Now, draw arrows between the boxes, with each arrow labeled with a probability. This network is your state space and transition matrix. The arrows dictate how your system moves from one state to another, like a game of musical states.
Initial Conditions: The Starting Point
Just like a journey has a starting point, Markov chains need initial conditions. This is the probability distribution that sets the stage for your system’s adventures. It determines how likely it is to start in each state. As the chain progresses, these probabilities evolve, shaping the path ahead.
Properties of Markov Chains: The Good, the Bad, and the Intriguing
Now, let’s chat about the defining features of Markov chains. They come with a bag of tricks, like:
- Ergodicity: Is your chain a restless traveler, visiting every state eventually?
- Recurrence: Do states keep making a comeback like an unstoppable boomerang?
- Aperiodicity: States don’t play by set schedules, avoiding predictability.
- Positive Recurrence: States get a limited-time pass to visit, like a sojourn with an expiration date.
Markov Property and Chapman-Kolmogorov Equations: The Math Behind the Magic
The Markov property is like a charm that says your future is blind to the past, only caring about the present. And the Chapman-Kolmogorov equations are the magic formulas that predict future probabilities based on present states. These tools are like the GPS of Markov chains, guiding their probabilistic journeys.
Wrapping Up: Markov Chains in Real Life
Markov chains aren’t just abstract concepts. They’re used in weather prediction, economics, and even genetics. They’re the secret sauce that helps us understand and predict the unpredictable, making them a powerful tool for unraveling the mysteries of probability.
Explain the Markov property: the future evolution of the chain depends only on the present state.
Unraveling the Mysteries of Markov Chains
Imagine you’re strolling through a park, lost in thought. Suddenly, a stranger approaches you and hands you a deck of cards. He asks you to flip over the top card, which happens to be an ace. Curious, you wonder what the next card will be. Could it be another ace?
That’s where Markov chains come in! They’re like those cards: what happens next in the sequence depends only on what’s happening now, not on the past. It’s like life itself: sometimes you have good days, sometimes bad, and the future is always a bit of a mystery.
State Space and Transition Matrix
Think of the park as the state space, where each card represents a state. And just like in real life, there are certain paths you can take from one state to another. That’s where the transition matrix comes in. It tells you the probability of moving from one state to another, like flipping an ace and getting a two.
Initial Conditions
Every journey starts somewhere. For Markov chains, it’s the initial probability distribution. It’s like the starting point of your walk in the park. And as you keep walking, you might notice that certain states appear more frequently than others. That’s the stationary distribution, which tells you the long-term behavior of the chain.
Properties of Markov Chains
Markov chains have some funky properties that make them extra special. They can be ergodic, meaning they eventually visit all states. They can also be recurrent, where you keep coming back to certain states like a magnet. Or asymptotically stable, like a seesaw that eventually settles into a balanced position.
Markov Property
Here’s the key: the Markov property. It’s what makes these chains so cool. It says that the future of the chain only depends on the present state. It’s like having a secret map that tells you what’s next without knowing the whole path. And that’s what makes Markov chains so useful in predicting things, from weather patterns to the stock market!
Demystifying Markov Chains: A Beginner’s Guide to State-Space Shenanigans
Yo, Markov Chains! Meet the Party Rockers of Probability
Imagine a magical realm where your next move is solely dictated by your present state, and where randomness and patterns dance hand in hand. That’s the wild world of Markov chains, the ultimate party rockers of probability!
State Your Case: The State Space and Transition Matrix
Think of a Markov chain as a party with a bunch of different rooms (states). You can hop from one room to another, and the rules for this merrymaking are defined by a transition matrix. This matrix shows you the odds of moving from one room to any other room on the dance floor. It’s like a roadmap for your random adventures!
The First Step: Initial Conditions
Every party needs a starting point, right? For Markov chains, that’s where the initial conditions come into play. They tell us where you’re starting from so we can calculate your odds of ending up in any other room as the party rages on.
Party Time: Properties of Markov Chains
Now, let’s talk about the party dynamics! Markov chains have some funky properties:
- Ergodicity: Can you hit up all the rooms eventually?
- Recurrence: Do you keep coming back to the same old rooms?
- Stability: Does the party hit a steady groove or does it keep jumping around?
- Irreducibility: Can you move between any two rooms without getting stuck?
- Aperiodicity: Do you visit rooms in a predictable pattern or randomly?
- Positive Recurrence: Do you always have a chance of getting back to a room eventually?
The Markov Property and Chapman-Kolmogorov Equations: Double Trouble
Markov property is like the golden rule of the party: your next move only depends on your current room. The Chapman-Kolmogorov equations are the mathematical equations that let us figure out the odds of moving between rooms based on the transition matrix and the initial conditions. They’re like the DJs who mix the beats and keep the party going!
