# Different types of sample spaces in probability

In probability, sample space is a set of all possible outcomes of an experiment.

A sample space can be finite or infinite.

A sample space can be discrete or continuous.

A sample space can be countable or uncountable.

From some texts I got that finite sample space is same as discrete sample space and infinite sample space is continuous sample space.

But some texts are saying that countable sample space is discrete sample space and uncountable sample space is continuous sample space.

Which one of the following above is correct?

I got confuse because of the following two statements in this text book

Discrete Probability Law :

If the sample space consists of a finite number of possible outcomes, then the probability law is specified by the probabilities of the events that consist of a single element. In particular, the probability of any event $\{s_1, s_2, . . . , s_n\}$ is the sum of the probabilities of its elements.

Continuous Models :

Probabilistic models with continuous sample spaces differ from their discrete counterparts in that the probabilities of the single-element events may not be sufficient to characterize the probability law.

Discrete probability law deals with finite sample spaces, but continuous probability models deal with continuous sample spaces. So I am confused whether countably finite sample spaces comes under which probabilistic model.

listen this also for accuracy.

• These are definitions. The second one seems to be the usual definition. – zoli Oct 22 '17 at 11:40

## 1 Answer

The sample space of an experiment may consist of a finite or an infinite number of possible outcomes.

Finite sample spaces are conceptually and mathematically simpler. Still, sample spaces with an infinite number of elements are quite common. As an example, consider throwing a dart on a square target and viewing the point of impact as the outcome.

For the case discrete probabilities there are two possible finite and infinite sample space.

1. case throw the two dice just for once, is example of finite sample space, because it's only consist of sample space between ((1,1),(1,2),...,(6,6)) and it's countable.

2. case throw two coins is another example of finite sample space with sample space (HH,HT,TH,TT), sure it's countable.

3. case probability on how much i have to throw the two coins to get result HH, then this is an example of discrete probability with infinite sample space, we maybe just one throw in order to meet the condition or maybe we need 1000 experiment to get the result HH, so the number of toss may vary and had it's own probability. You could refer to this lecture and it's countable.

4. case continuous probability, like we throw the dart, this is an example of infinite sample space and also uncountable too.