Joint Probability $P(X,Y,Z) = P(Y,X,Z)$ Does the order of variables in the joint probability $P(X,Y,\dots)$ have any implications on the statement of joint probability? Concrete, is:
$$P(X,Y,Z) = P(Y,X,Z)$$
To my mind, clearly this is correct, can someone assure me?
 A: First, let us use canonical notations: for every random variable $U$ with values in $\mathbb R^n$, $\mathrm P_U$ is the measure on $\mathcal B(\mathbb R^n)$ defined by $\mathrm P_U(B)=\mathrm P(U\in B)$ for every $B$ in $\mathcal B(\mathbb R^n)$. Hence your question is whether, for every real valued random variables $X$, $Y$ and $Z$ defined on the same probability space and every $B$ in $\mathcal B(\mathbb R^3)$,
$$
\mathrm P((X,Y,Z)\in B)=\mathrm P((Y,X,Z)\in B)\ ?
$$
In particular, one would have
$$
\mathrm P(X\in B,Y\in C)=\mathrm P(Y\in B,X\in C).
$$
Dubious, don't you think?
Edit Here is an example. Assume that $X$ and $Y$ are independent, that $X$ is Bernoulli with parameter $p$, hence $\mathrm P(X=1)=1-\mathrm P(X=0)=p$, and that $Y$ is geometric with parameter $a$, hence $\mathrm P(Y=n)=(1-a)a^n$ for every integer $n\geqslant0$. Choose $B=\{0\}$ and $C=\{1\}$. Then,
$$
\mathrm P(X\in B,Y\in C)=(1-p)(1-a)a,\qquad \mathrm P(Y\in B,X\in C)=(1-a)p.
$$
Second edit Let us recall that, if $X$ is a random variable with values in, say, a discrete space $E$, then $\mathrm P_X$ is the unique distribution on $(E,2^E)$ such that, for every $B\subseteq E$, $\mathrm P_X(B)=\mathrm P(X\in B)$. Likewise, if $X$ and $Y$ are random variables (defined on the same probability space) with values in some discrete spaces $E$ and $F$, then $\mathrm P_{(X,Y)}$ is the unique distribution on $(E\times F,2^{E\times F})$ such that, for every $B\subseteq E\times F$, $\mathrm P_{(X,Y)}(B)=\mathrm P((X,Y)\in B)$.
Thus, three probability measures are involved: $\mathrm P$ is a probability measure on the probability space (usually denoted) $\Omega$ (and in fact one never uses $\Omega$ nor $\mathrm P$ to perform computations), $\mathrm P_X$ is a probability measure on $(E,2^E)$, and $\mathrm P_{(X,Y)}$ is a probability measure on $(E\times F,2^{E\times F})$.
Recall finally that a probability measure is a function defined on a collection of subsets of a given set, with values in $[0,1]$. For example, $\mathrm P_X:2^E\to[0,1]$. The images $\mathrm P_X(B)$ for $B\subseteq E$ are real numbers in $[0,1]$, not $\mathrm P_X$ itself.
