How am I begging the question? That isn't clear.
I don't believe that the probability of life in our universe is almost zero, but you can read all about the Fine-Tuned Universe argument at Wikipedia which is how this claim is most often made. My post is not targeted at the question of whether life really is that unlikely (I suspect that the parameters needed for life are not as strict as often claimed, only for life as we know it). The question I am trying to examine is whether the anthropic principle is a sufficient counter to the Fine-Tuned Universe (it is not); I'll leave it to other people to discuss other objections to the Fine-Tuned Universe.
The Fine-tuned Universe Theory, according to Wikipedia is the belief that, "our universe is remarkably well suited for life, to a degree unlikely to happen by mere chance". It is typically used to argue that our universe must therefore be the result of Intelligent Design.
One of the most common counter-arguments to this view based on the Anthropic Principle. The argument is that if the conditions were not such that life would be possible, then we would not be able to observe this, as we would not be alive. Therefore, we shouldn't be surprised that the universe has favourable conditions.
I am going to argue that this particular application of the anthropic principle is in fact an incorrect way to deal with this problem. I'll begin first by explaining one way to deal with this problem; afterwards I will explain why the other way is incorrect.
Two model approach
We begin with two modes:
However, this is actually asking the wrong question. It is right to note that we shouldn't be surprised to observe that we survived given that it would be impossible to observe otherwise. However, if we were then informed that we lived in a normal, unbiased universe, rather than in an alternate biased universe, if the maths worked out a particular way such that it leaned heavily towards the alternate universe, then we would be surprised to learn we lived in a normal universe. In particular, we showed how this could work out above, when we examined the situation where p(we exist|normal universe) approached 0. The anthropic argument against the alternate hypothesis denies that surprise in a certain sense can occur, however, if fails to show that surprised in another, more meaningful sense can occur.
=p(we exist|normal universe)p(normal universe) + 1 - p(normal universe)
Performing Bayesian updates
Again, we'll imagine that we have a biased universe where we have 100% chance of being alive.
We will use Bayes law:
p(a|b)=p(b|a)p(a)/p(b)
Where:
a = being in a normal universe
b = we are alive
We'll also use:
p(alive) = p(alive|normal universe)p(normal universe) + p(alive|biased universe)p(biased universe)
Example 1:
Setting:
p(alive|normal universe) = 1/100
p(normal universe) = 1/2
The results are:
p(we are alive) = (1/100)*(1/2)+1*(1/2) = 101/200
p(normal universe|alive) = (1/100)*(1/2)*(200/101) = 1/101
Example 2:
Setting:
p(normal universe)=100/101
p(alive|normal universe) = 1/100
p(normal universe) = 100/101
The results are:
p(we are alive) = 100/101*1/100+1/101*1 = 2/101
p(normal universe|alive) = (1/100)*(100/101)* (101/2) = 1/2