# Outcome Space and Sample Space

Hi, I am new to this forum.

I was wondering if someone could explain to me what is the difference between the outcome space and the sample space. My teacher gave me the definition but I can't seem to understand what is the difference. He also gave me an example but I need help with it too.

Outcome space is the set of elements that are the possible outcomes for the experiment
Sample space is a set of points for which outcomes can be described.

He also gave me an example: Suppose we bought a bag of apples from a farmer. The outcome space is the set whose elements are the farmer's bags of apples available for our purchase. The sample space is the same as the outcome space. I understand what the outcome space is but why is the sample space the same as the outcome space ? Can someone give me an example when the sample space is different from the outcome space?

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Well this is the first time I've used a forum like this to answer a question so here goes ...

The best way for me to explain this is to give an example. Let's use the example of rolling one die. The sample space is {1, 2, 3, 4, 5, 6}, basically a list of the possible result you get from rolling a die. The outcome space is the list of possible outcomes, i.e. the possible results from the experiment. If that experiment is to role the die once, the sample and outcome spaces are the same as you are sampling once from the sample space (i.e. rolling the die).

If the experiment is to role the die twice however, then let (a, b) be an outcome where a is the result of the first roll, and b the result of the second roll. Your sample space is still {1, 2, 3, 4, 5, 6} (i.e. what a, and b could be) but your outcome space is {(1,1), (1,2), ..., (1,6), (2,1), (2,2), ..., (2,6), ..., (6,1), (6,2), ..., (6,6)} i.e. there are 36 outcomes, each resulting from us sampling twice from the sample space (and allowing us to have the result of the first sample (roll) occurring again in the second sample).

In summary, the sample space lists the possible results of each part your experiment (e.g. each roll of the die). The outcome space is the list of possible outcomes, which is the result of the whole experiment (e.g. in the second sample the two numbers we get from rolling a die twice). And it is possible that the outcome space is identical to the sample space.

I hope this clears things up for you mate!

Hi lynniiee! I think your teacher is strange. I have always encounter the words "outcome space" and "sample space" as the same thing.

And the description that he gives isn't very clear either. Can you perhaps provide a reference book or something, so I can see what he actually means...

Stephen Tashi
Perhaps the teacher wishes to distinguish between the possible results of an underlying probability model for a phenomena and the possible measurements of them that we can observe. For example, if the farmer's apple's are put in boxes and a shipping company measures the weights of the boxes then we might form some probability model for the weights and volumes of the individual apples (the sample space) and this would imply (considering the volume of a box) the possible outcomes for the weights of the boxes, which would be an outcome space. This is along the line of what StatsTiger suggests. However, I agree wtih micomass that this terminology isn't commonly used in English probability texts and I'm just trying to mind read your instructor.

I don't understand what your teacher means either.

Here is a site that treats "sample space" and "outcome space" as synonyms. This seems natural, given that Wikipedia: Probability space defines "outcomes" as elements of (=points in) the sample space. Other sources define "outcome space" exactly as Wikipedia defines "sample space", as the set of all possible outcomes:

"The set of all possible outcomes is called the outcome space (sometimes the sample space)"

"The outcome space is the set of all possible outcomes of a given random experiment. The outcome space is often denoted by the capital letter S." (There's no entry in this glossary for "sample space".)

The impression I get from Googling is that there is no standard distinction between the two terms. All of the sources I've checked so far treat them as synonymous, and many explicitly state that they mean the same.

StatsTiger's attempt to give distinct meanings to the terms "sample space" and "outcome space" leaves me with a lot of questions. What is the general distinction? Is "outcome space" defined for all probability spaces? If so, how? If not, what kind of probability space does it apply to: only ones describing one or more die rolls, only ones involving dice or apples, only finite ones, only countable ones,...?

Other uses. In this Glossary of game theory, outcome space is said to be identical "in most textbooks" to "Rn, describing how much gain (money, pleasure, etc.) the players are allocated by the end of the game."