# 1. Basic Probability Concepts

Def 1.1: The sample space is the set $S:=\{\text{all elements that are in 1-1 correspondence with the set of all outcomes of an experiement}\}.$

Questions…

1. What exactly does this mean?
2. How is this different from $S_1:=\{\text{the set of all outcomes of an experiment}\}$? The “1-1 correspondence” part makes me think that $S_2:=\{f:X\longrightarrow\{\text{all outcomes of an experiment}\}\;|\;f\text{ is one-to-one}\}$, where $f$ corresponds with the term “element” used in Def 1.1.

Ex: Experiment=tossing 3 fair coins
$S=\{HHH, HHT, HTH, THH, TTT, THH, TTH, THT\}$

I think this corresponds with my $S_1$, at least in the way that I’m thinking about $S_1$. Perhaps the difference is that I’m thinking that there is only one way to represent the set of all outcomes when, in fact, it could be represented in multiple ways.

Ex: $S^{(1)}=\{000,001,010,100,111,100,110,101\}$
$S^{(2)}=\{O_1,O_2,O_3,O_4,O_5,O_6,O_7,O_8\}$, where $O_1$ corresponds with $HHH$, $O_2$ corresponds with $HHT$, etc. So all of these are “equivalent” representations of $S$.

If that is the case, then I think that the definition of $S$ should be modified.

Def 1.1′:sample space of an experiment is a set $S:=\{\text{elements that can be put into 1-1 correspondence with the set of all outcomes of the experiment}\}$ or $S:=\{X\;|\;\text{there exists a 1-1 function }f:X\longrightarrow\{\text{all possible outcomes of the experiment}\}\}$.

• I added the phrase “an experiment” to specify that a sample space corresponds with a particular experiment”.
• I changed “the sample space/set” to  “a sample space/set” since there could be multiple representations

Questions…

1. I wonder if my incorporation of representation is problematic. For example, does my definition now imply that $S\neq S_1$, or does it allow for isomorphisms? Is there any value to viewing them as different? Or should $S:=\mathbb{S}=\{S\text{ that satisfy Def 1.1'}\}$?How are these different? How are they the same?

I just looked up the definition of sample space in A First Course in Probability (Ross, 2006), and this is what I found:

“This set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $S$” (p. 24).

This definition corresponds with my $S_1$.

Citation
Ross, S. (2006). A First Course in Probability. (7th ed.) Pearson Education: Upper Saddle River, NJ.