What is the a priori probability that we are here? It is interesting how the problem breaks into three pieces.
We can write an equation for the probability that we exist1:
Pwe exist = Psomething instead of nothing x Nplanets x Pplanet supports life x Plife evolves
The first piece is Psomething instead of nothing is the probability that there is something instead of nothing. This has been a point of great contention, with theologians and some philosophers declaring that it is absurd that nothing can produce something, and physicists countering that the laws of quantum mechanics permit it. Arguing over this factor is pointless. Might as well concede that it has a value of 1.0.
The third piece, Plife evolves, the probability that life evolves, is also hotly debated. There is much to attack in evolution, and virtually every scientific discovery, apart from witnessing an actual speciation, will stress rather than support evolution. Why is that? Well, evolution needs time. More than anything, it would like it to be that complex life does not appear until late in the game. The problem is the earth's crust was molten to about 3.9 billion years ago, and fossils of cells have been dated to 3.5 billion years. Limestone (formed by dead organisms) has been dated to 3.8 billion years ago, and ancient sediments even older. So evolution, instead of having a billion or so years to produce life, may have only 100 million years or less. Any future discovery cannot give evolution more time, only less. Compound this by the fact that investigations into the biochemistry of single cell creatures continues to reveal more complexity, not less. There is rarely a good news day for the evolutionists.
However, the interesting game to me is the middle term. If evolution can be likened to a tornado assembling a 747 from scrap in a junkyard, the middle term, ignored by biologists, is the probability that right scrap just happens to be lying about. For years, the large value for the number of planets in the universe, Nplanets, which is about 1022, gave people great confidence that the number of earth-like planets must be large. SETI receives funding because 1022 is a large number. Pplanet supports life, the probability that a planet can support life, was ignored. After all, even if it were one in a trillion, that would still leave 1010 earth-like planets.
The amazing development is that it looks like Pplanet supports life is not 1 in a trillion, or 10-12, but more like 10-180! (This sources argues that it is even much smaller that that, more like 10-215! ) This makes the likelihood of even one earth-like planet, arising accidentally, to be vanishingly small.
How small is 10-180? There are ~ 1019 grains of sand on earth. Suppose that is typical. There are ~ 1022 planets. With these assumptions, there are ~ 1041 grains of sand in the universe. Suppose there are four special grains of sand hidden anywhere in the universe. The chance to pick randomly four grains of sand anywhere in the universe and get the right four is about 1 in 10164. This is still far more likely that the possibility of an earth-like planet.
The mathematically astute will notice that we approximated [1 - (1 - P)N] as N x P, a very accurate approximation when P is small.