Complexity is not an argument against randomness.
And I think you may be confusing prior probabilities with conditional probabilities. You're interested in the probability of life given some observed data (such as DNA), not just the prior probability of observing the data. You're interested in this question:
L=life
O=Observed data
P(L|O)=P(O|L)P(L)/P(O)
Essentially, I think you need to show that given the observed data, the probability of life is low, then you would need to that a rare observation wasn't due to random chance.
How did you estimate those odds? Calculating probabilities and likelihoods is not always intuitive. In the Milky Way alone there are estimated to be about 9 billion earth-like planets that live in the "goldilocks zone." Sure, if you estimate the odds of an earthlike planet bearing life and found it to be high, then it might be difficult to say that it was a statistical anomaly. And the question you are interested in isn't simply the probability of life, you are interested in the probability of random chanced life vs God created life. If you were to calculate the probability of an earthlike planet bearing life, and you found that the probability was remote, you will still have some sort of statistical error term where the observation of an earthlike planet bearing life would occur by random chance. You are interested in the hypothesis that life on THIS earth (or perhaps life on ANY earthlike planet or even all earthlike planets) was created by God vs the alternative hypothesis that it wasn't. I don't know how one would actually begin to try to demonstrate this; I really doubt it could be done. I think you would have to at least begin with an axiomatic statement that God exists, which usually undermines the initial purpose of the line of inquiry.
One of the major problems related to the above is that an observational piece of data cannot statistically establish causality, probabilities, or likelihoods. My observation of X says nothing about the cause of X, the population, the distribution of observations, the variance of observed data, etc...
Somewhat separately, I don't know that there can be random chance in a theistic worldview. You could never [statistically] demonstrate a God-caused event vs. a random event without a sufficient sample of God caused events in your data from which you could make statistical comparisons - and I doubt anyone has that data without assuming God's existence as axiomatic. Without that data, a God-caused event will appear no different from a random event. Essentially, God would be a statistical error term.
Or, if your observed data indicated that the error term was correlated with the dependent variable, without the data of God-caused events, God-causation would appear as omitted variable bias, which isn't that much different from an atheist saying, "oh, it's God of the gaps."
So I think when talking probabilities and likelihoods, without a prior commitment to belief in God, or sufficient data of God-actions, God-caused explanations would appear no different from random chance or omitted variable bias. For the theist, I think God has to (a) be an axiomatic belief and (b) leave/provide/grant/give sufficient data from which you could make further statistical comparisons.
Bottom line, I doubt statistics is very useful for theistic proofs.
And I think you may be confusing prior probabilities with conditional probabilities. You're interested in the probability of life given some observed data (such as DNA), not just the prior probability of observing the data. You're interested in this question:
L=life
O=Observed data
P(L|O)=P(O|L)P(L)/P(O)
Essentially, I think you need to show that given the observed data, the probability of life is low, then you would need to that a rare observation wasn't due to random chance.
How did you estimate those odds? Calculating probabilities and likelihoods is not always intuitive. In the Milky Way alone there are estimated to be about 9 billion earth-like planets that live in the "goldilocks zone." Sure, if you estimate the odds of an earthlike planet bearing life and found it to be high, then it might be difficult to say that it was a statistical anomaly. And the question you are interested in isn't simply the probability of life, you are interested in the probability of random chanced life vs God created life. If you were to calculate the probability of an earthlike planet bearing life, and you found that the probability was remote, you will still have some sort of statistical error term where the observation of an earthlike planet bearing life would occur by random chance. You are interested in the hypothesis that life on THIS earth (or perhaps life on ANY earthlike planet or even all earthlike planets) was created by God vs the alternative hypothesis that it wasn't. I don't know how one would actually begin to try to demonstrate this; I really doubt it could be done. I think you would have to at least begin with an axiomatic statement that God exists, which usually undermines the initial purpose of the line of inquiry.
One of the major problems related to the above is that an observational piece of data cannot statistically establish causality, probabilities, or likelihoods. My observation of X says nothing about the cause of X, the population, the distribution of observations, the variance of observed data, etc...
Somewhat separately, I don't know that there can be random chance in a theistic worldview. You could never [statistically] demonstrate a God-caused event vs. a random event without a sufficient sample of God caused events in your data from which you could make statistical comparisons - and I doubt anyone has that data without assuming God's existence as axiomatic. Without that data, a God-caused event will appear no different from a random event. Essentially, God would be a statistical error term.
Or, if your observed data indicated that the error term was correlated with the dependent variable, without the data of God-caused events, God-causation would appear as omitted variable bias, which isn't that much different from an atheist saying, "oh, it's God of the gaps."
So I think when talking probabilities and likelihoods, without a prior commitment to belief in God, or sufficient data of God-actions, God-caused explanations would appear no different from random chance or omitted variable bias. For the theist, I think God has to (a) be an axiomatic belief and (b) leave/provide/grant/give sufficient data from which you could make further statistical comparisons.
Bottom line, I doubt statistics is very useful for theistic proofs.