Can we use probability or information theory to conclude whether a complex phenomenon like life would be likely or unlikely to arise randomly?
I'm not religious nor have I ever been but it seems astonishing to me that all of the complexity of life and human consciousness would evolve naturally. Typically proponents of Intelligent Design are religious conservatives with general beliefs against science and in favor of a Christian God which leads it to be dismissed as an attempt to shutdown scientific debate and discovery, or from the POV that a moral God wouldn't code for useless regions of DNA or harmful mutations. I agree the Theory of Evolution makes good predictions that have advanced science but I was wondering if there's a way to estimate whether a process as complex as evolution and structure as informationally complex as life can just randomly arise or not (in which case philosophical arguments about living in the matrix or God-driven evolution must be considered).
If I flip a coin and I expect it to be unbiased or biased in a specific way I can calculate the cross entropy of the expected probability and the observed frequencies of heads and tails to conclude how surprised I am and how likely it is the coin behaves as I see. Similarly from what I understand in medicine a null hypothesis is made, say that the drug doesn't work, and if the results are such that they would be extremely unlikely given a normal distribution that follows the null hypothesis it can be dismissed.
Is there a similar way to simulate early conditions on earth and see how likely it is that life would arise (in how many simulations under random conditions it arises)? Or to start with simple one-cell organisms and see how likely it is for far more complex life forms to evolve in a computer simulation? Or at least information-wise, say there's XYZ amount of information contained in the human DNA in a very specific order for humans to exist, how likely is that if the formation of the universe/Earth is largely a random process?