The answers already given amount to, "With probability 1, a probability-zero event will happen when you do something randomly." This is absolutely correct, but let me give a slightly different take on the OP's argument that is more specifically about well-orderings.

Let's simplify matters, and just do the following: select $\alpha,\beta\in\omega_1$ randomly. What is the probability that $\alpha<\beta$ (in the usual ordering on $\omega_1$)? Naive arguments show that the probability is 1. But by those same arguments, the probability that $\beta<\alpha$ is 1.

In the reals context, the paradox is: if I select a real $r=\langle s_0, s_1\rangle$, what is the probability that $r\in X$, where $$ X=\lbrace u=\langle v_0, v_1\rangle: f^{-1}(v_0) < f^{-1}(v_1)\rbrace,$$ and in turn $$ f: \omega_1\rightarrow\mathbb{R} $$ is a bijection? (Note that this argument is really about well-orderings, and not just about the inevitability of probability-zero events.)

This is just Freiling's argument against $CH$ (see http://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry). This argument was discussed on MO here: Axiom of Symmetry, aka Freiling's argument against CH. Briefly, the reason it isn't generally found convincing as an argument against $CH$ is that it tacitly assumes that the reals are well-orderable if and only if they are well-orderable in a measurable way. So if you believe that there are non-measurable sets, this argument really shouldn't be very convincing.

(On the other hand, if you find this argument intuitively appealing, maybe $L(\mathbb{R})$ is right for you!)

The answers already given amount to, "With probability 1, a probability-zero event will happen when you do something randomly." This is absolutely correct, but let me give a slightly different take on the OP's argument that is more specifically about well-orderings.

Let's simplify matters, and just do the following: select $\alpha,\beta\in\omega_1$ randomly. What is the probability that $\alpha<\beta$ (in the usual ordering on $\omega_1$)? Naive arguments show that the probability is 1. But by those same arguments, the probability that $\beta<\alpha$ is 1.

In the reals context, the paradox is: if I select a real $r=\langle s_0, s_1\rangle$, what is the probability that $r\in X$, where $$ X=\lbrace u=\langle v_0, v_1\rangle: f^{-1}(v_0) < f^{-1}(v_1)\rbrace,$$ and in turn $$ f: \omega_1\rightarrow\mathbb{R} $$ is a bijection? (Note that this argument is really about well-orderings, and not just about the inevitability of probability-zero events.)

This is just Freiling's argument against $CH$ (see http://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry). This argument was discussed on MO here: Axiom of Symmetry, aka Freiling's argument against CH. Briefly, the reason it isn't generally found convincing as an argument against $CH$ is that it tacitly assumes that the reals are well-orderable if and only if they are well-orderable in a measurable way. So if you believe that there are non-measurable sets, this argument really shouldn't be very convincing.

(On the other hand, if you find this argument intuitively appealing, maybe $L(\mathbb{R})$ is right for you!)