So long as the chances of beneficial mutation are non-zero, the probability will be non-zero. That's what the math says.
You also have to take into account the exponential growth of populations. The non-zero chance of a beneficial mutation is multiplied by the number of organisms available. This number grows rapidly.
So you have exponential growth combined with non-zero probabilities and over a billion years to work with.
The math may not "support" the notion of change from microbe to man, but it certailly doesn't refute it, either.
Try again "math whiz".
Okay. Here is a non-zero probabi;ity and what it means in Real world terms.
If you were to attempt to assemble one short prtein of 100 amino acids by random chance (which controls in the case of neutral mutations) it goes like this:
1. all peptide bonds. 50-50 chance in nature. 1 chance in 2^100 or 1in 10^30.
2. All left handed amino acids. Again 50-50 so 1 chance in 10^30.
3. 50% of the amino acids mission critical. Out of 20 amino acids. 1 chance in 20^50 or 10^65.
4. 10^30 x 10^30 x 10^65 = 10^125 or 1 chance in 10^125 of randomly producing a short protein used by life.
5. This is a non-zero probability and seems reachable.
Let's try.
1. Suppose we assembled 1 protein per second since the beginning of time. 13.7 billion years = 4.3 x 10^16 seconds or let's just call it 10^17 seconds. We would need to compare 10^108 proteins per second to get ther.
2. Let's get Blue Gene, IBM's supercomputer to help. It operates at 70 teraflops or 7 x 10^13 floating point operations per second or we'll call it 10^14. So if we had Blue Gene woking since the beginning of time we would still fall a little short.
3. We need more computers. The number of photons in the Universe (per Penrose) is around 10^88. If every photon (10^88) was a blue Gene (x 10^14) operating since the beginning of time (x 10^17) you would still need a million universes fulll of computers to make it all work out.
So non-zeo probability ain't all its cracked up to be.