Determining the states in Markov chains

[stripes]

Homework Statement

This is not a homework question, just me trying to wrap my head around things. My probability class talked about Markov chains for less than 2 hours worth of lecture, and I've been super sick lately, so I'm still a little confused.

If we're considering real world examples where the probability of an event happening tomorrow depends on what happens in the previous n days, where each day, m different events could have happened, is the number of states just mⁿ?

Two practice problems that were given to us were as follows:

Joe's mood depends on his mood from the previous day only (so we're going back the previous 1 day). He could be gloomy, so-so, or cheerful (3 possible events). This means we have a Markov chain with 3¹ states?

The probability that it rains tomorrow depends on whether it rained the past 2 days (going back the previous 2 days). If the weather on any day can either be rainy or dry, then the number of possible events is 2. So the number of states is 2² = 4?

One more example I came across was the weather one but with more days: suppose the probability that it rains tomorrow depends only on whether it rained the past 3 days (going back the previous 3 days). It can either be rainy or dry (2 possible events). So the number of states is 2³ = 8?

I know that these answers are correct but is my way of thinking about right? I will be given such real world examples (involving previous days and what not) on my final exam, so when I'm trying to figure out the number of states in these real world examples, am I doing it correctly?

I guess I could just figure it out by going RRR (rain rain rain), RRD (rain rain dry), RDR, RDD, ... etc. I get this feeling that I'm totally missing the point (missing how to figure things out intuitively). I take it it's because I've missed some class due to illness.

If someone could let me know if I'm going about this correctly I would really appreciate it.

Homework Equations

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The Attempt at a Solution

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[Ray Vickson]

Homework Statement

This is not a homework question, just me trying to wrap my head around things. My probability class talked about Markov chains for less than 2 hours worth of lecture, and I've been super sick lately, so I'm still a little confused.

If we're considering real world examples where the probability of an event happening tomorrow depends on what happens in the previous n days, where each day, m different events could have happened, is the number of states just mⁿ?

Two practice problems that were given to us were as follows:

Joe's mood depends on his mood from the previous day only (so we're going back the previous 1 day). He could be gloomy, so-so, or cheerful (3 possible events). This means we have a Markov chain with 3¹ states?

The probability that it rains tomorrow depends on whether it rained the past 2 days (going back the previous 2 days). If the weather on any day can either be rainy or dry, then the number of possible events is 2. So the number of states is 2² = 4?

One more example I came across was the weather one but with more days: suppose the probability that it rains tomorrow depends only on whether it rained the past 3 days (going back the previous 3 days). It can either be rainy or dry (2 possible events). So the number of states is 2³ = 8?

I know that these answers are correct but is my way of thinking about right? I will be given such real world examples (involving previous days and what not) on my final exam, so when I'm trying to figure out the number of states in these real world examples, am I doing it correctly?

I guess I could just figure it out by going RRR (rain rain rain), RRD (rain rain dry), RDR, RDD, ... etc. I get this feeling that I'm totally missing the point (missing how to figure things out intuitively). I take it it's because I've missed some class due to illness.

If someone could let me know if I'm going about this correctly I would really appreciate it.

Homework Equations

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The Attempt at a Solution

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Your 8 states RRR, RRD, ... are correct; that is the way these things are ususally done.