Aren't they both the same? According to me they are just different terminologies used for different types of systems. What I understand is that sample space is used when we talk about static systems and state space when we talk about dynamic systems. Am I correct? If not then what exactly is the difference between them?
My teacher had asked for the state space in Markov Chain questions, so I'm guessing that state space is used only in dynamic systems? She said that if we roll 2 dice then the sample space will be {1,2,...,6},
but the state space will be {(1,1),(1,2),...,(6,6)}.
But as far as I remember, the sample space for rolling 2 dice is{(1,1),(1,2),...,(6,6)}.
3 Answers
$\begingroup$"State space" is used in lots of areas of math. For instance, "all possible positions for a particle, along with all possible velocities at those points" might be a state space for analyzing particle motion, or "all the possible pairs generated by rolling two dice" might be a state space for a gambling question. (see edit below)
"Sample space" usually means that in addition to a set of states there is also a "probability function" associated with the state space, which gives you a way to say which states are more or less likely to occur.
So samples spaces are examples of state spaces, but there are a lot more different types of states spaces than just sample spaces. However, if you're only taking Stats you can expect them to be used interchangeably.
Edit: According to Wikipedia, my example of "all possible positions ..." should be called a "phase space", because it's continuous, and state space is only proper for discrete spaces, but I'm pretty sure I've heard "state space" used in both discrete and continuous instances.
$\endgroup$ 2 $\begingroup$(Apologies for posting a second answer, but after re-thinking this question I think I've got a much better answer that's independent of my first one.)
"State space" is used in dynamics. It implies that there is a time progression, and that your system will be in different states as time progresses. (And Wikipedia says that "state space" implies that we're dealing with discrete time, and that "phase space" should be used in continuous time systems. I don't think that's required, but I'm not Wikipedia.)
"Sample space" hints very very heavily that we're going to be talking about probabilities, i.e. that there will be one (or more) probability measure functions defined on our sample space. (Technically they are defined on "$\sigma$-algebras", which are sets of events in our sample space satisfying certain properties, but the distinction is unimportant in discrete sample spaces.)
In the original question they referred to the "state space" of rolling two dice. This sounds wrong, because there is no dynamic evolution through time. What we have there is a list of all possible outcomes or events, concepts that go along with the term "sample space".
But let's look at an example of a simple Markov chain. Suppose it has states $A$, $B$, and $C$, along with some transition probabilities between the states that define how it evolves through time. Its state space is $\{A, B, C\}$. We can define different sample spaces based on this, like {all triples of first three states} or {all infinite sequences of states}. These are distinct sample spaces, and each has their own probability distribution. So Markov chains can lead to confusion as we could apply either term to them. Heck, if we were discussing {all initial states}, then our sample space would be $\{A, B, C\}$, which is identical to our state space, the only difference being that if I say "sample space" we'd expect to be thinking about their probabilities, unlike "state space", which doesn't carry that connotation.
$\endgroup$ 2 $\begingroup$The sample space is the set that contains all possible outcomes (events) for an experiment, and is usually denoted by $\Omega$ (the set of all possible events $\omega$).
For example, when tossing two coins, $\Omega=\{\text{(heads,heads)},\text{(heads,tails)},\text{(tails,heads)},\text{(tails,tails)}\}$
As pointed out by user275313, this answer was not completely right. The state space is the set of all possible states in which a dynamic system can be, and it denotes that the system evolves continuously from one to the next.
On the other hand, the state space refers to the set of all possible realizations of a random variable. Remember that a random variable is a function that maps events (abstract) to mathematically representable entities (e.g. numbers).
For example, let $X$ be a random variable associated with tossing a coin, so that:
$X(\omega )={\begin{cases}1,&{\text{if}}\ \ \omega ={\text{heads}},\\0,&{\text{if}}\ \ \omega ={\text{tails}}.\end{cases}} $
Therefore, if $Y$ is a multivariate random variable that for an experiment of tossing two coins, $Y=\{X_1,X_2\}$, the state space will be $Y=\{(1,1),(1,0),(0,1),(0,0)\}$.
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