What is the probability of 3 possible products, between two chemical species?
Well, if you assume the rates are known and the reactions' order follows from stoechiometry (e.g. if they are elementary reactions), you can put the chemical kinetics into simple equations:

$$\frac{\mathrm da}{\mathrm dt} = -k_1 a(t)^2 - k_3 a(t)b(t) $$
$$\frac{\mathrm db}{\mathrm dt} = -k_2 b(t)^2 - k_3 a(t)b(t) $$

($t$ here being time, not temperature).

Knowing initial amounts or concentrations $a_0=a(t=0)$ and $b_0=b(t=0)$, you can pretty much integrate  the system to find out what happens.



Edit — solving this system for $k_1=k_2=k_3$ yields the quantities of AA, AB and BB at infinite time to be as follows:

$$aa = \frac{a_0^2}{a_0+b_0}$$
$$ab = \frac{a_0 b_0}{a_0+b_0}$$
$$bb = \frac{b_0^2}{a_0+b_0}$$