Assuming we stabilize humanity's genetic code and prevent further modification (whether by direct manipulation or eugenics) and further assume we do not intentionally deviate from the "human template" we can do some neat math,
The earth probably can't house more than a trillion in even the most generous estimations without extreme advances and some think we're already nearing a max of 10 billion. lets assume a generation of 20 years. The solar system has a few places we can potentially manipulate into being habitable: mars, venus, the asteroid belt, and europa. Let's be generous and assume they can all support another trillion each. This balances with the 20 year generation for 1 trillion people born in the solar system every 20 years. Now, lowball estimates say there's about 10^24 stars, lets be generous and say 10^30 and that they all are capable of fitting similar amounts of people as the solar system. We're about 10^100 years from the heat death and assuming (incorrectly) that as stars die off we'll find replacements and we'd be able to keep this many people alive until the heat death.
This is $10^{30}\cdot\left(5\cdot10^{12}\right)\cdot\frac{10^{100}}{20}=2.5*10^{141}$ people in total. Compared with the $420\cdot10^{9}$ possible people this means (assuming, unrealistically a random
selection of people at birth from all possibilities) that the genetic description of a person would occur on average $\frac{\left(10^{30}\cdot\frac{\left(5\cdot10^{12}\right)}{5}\cdot\frac{10^{100}}{20}\right)}{\left(420\cdot10^{9}\right)} 1.1904761905×10^{129}$
times.Even swapping out for low ball stuff, 10 billion people per earth, one earth for every quadrillion suns etc, we get $\frac{\left(\frac{10^{24}}{10^{15}}\cdot\frac{\left(10\cdot10^{9}\right)}{5}\cdot\frac{10^{100}}{20}\right)}{\left(420\cdot10^{9}\right)}=2.38\cdot 10^{105}$ of you.
The time to conquer space is miniscule in the face of 10^100 years, so it is ignored.
Thoughts?