Is not evolution just too improbable to have occurred purely by natural processes? No, it is not.

Critics of evolutionary theory are fond of doing probability calculations, such as the assembly of some complex protein molecule from random amino acid sequences, coming up with a very small number, and then asserting that evolution is for all practical purposes, statistically impossible. There are at least three problems with this logic.

1. Often, 10^-50 or 10^-100 are invoked as probability thresholds of being so unlikely as to be statistically impossible. For single events, this may be the case, but for series of events, it is not. If you flip a coin 166 times, there are 2¹⁶⁶ possible sequences of heads and tails, and thus, the probability of whatever sequence of heads and tails you obtain is 2^-166, which is roughly 10^-50. Similarly, if you flip the coin 332 times, the probability of whatever sequence of heads and tails you obtain is 2^-332, which is roughly 10^-100. Performing such a simple exercise only takes a few minutes and will yield a highly improbable outcome; nevertheless, the fact that you do flip a coin 166 or 332 times, means that the probability of obtaining some sequence is 1. Similarly, if you shuffle a deck of cards and then lay them out in sequence, whatever sequence of 52 cards you lay out has a probability of 1/52!, which is roughly 10^-68. So improbable sequences of events happen all the time.

2. The probability that a rare event will occur at least once increases with the number of tries. An interesting special case of this phenomenon is the following. What is the probability of seeing a rare event, whose individual probability is 1 in N, occur at least once given N tries?

If the probability of obtaining the rare event in one try is 1/N, the probability of not obtaining the rare event in one try is (1-1/N). The probability of not seeing a rare event occur at least once in N tries is (1-1/N)^N.  Thus, the probability of seeing a rare event with probability 1/N occur at least once in N tries is...

[1-(1-1/N)^N].

The limit of this expression, as N approaches infinity, is (1-1/e) or roughly 0.63, where e is the base of natural logarithms. (Incidentally, e is defined as the limit of (1+1/N)^N as N approaches infinity.)

So if the probability of a rare event is 1/10⁵⁰, and you have 10⁵⁰ tries to achieve it, the probability of seeing that rare event at least once is ~0.63. As the number of tries increases, so does the probability of seeing that rare event. For example if you had double the number of  tries (i.e. 2N in our example), the probability of seeing the rare event occur at least once becomes...

[1-(1-1/N)^2N].

The limit of this expression as N approaches infinity is (1-1/e²), which is roughly 0.86.

As the number of tries increases, the probability of seeing a rare event occur at least once increases.

3. Mutation plus natural selection greatly increase the probability of evolution achieving a particular result, and in doing so, also reduce the number of tries and length of time required before a particular result is achieved. From the Richard Dawkins book, The Blind Watchmaker, there is the example of, 'what's the probability of monkeys writing Shakespeare by randomly banging on a typewriter?' Here's a link to website that illustrates this problem.

The sentence Dawkins uses is, "Methinks it is like a weasel" The sentence contains 28 characters (counting spaces), and out of 68 possible characters appearing at each point in the sentence, there are roughly 10⁵² possible sequences of characters. If we had 10⁵² tries, there would be a 0.63 chance of seeing that sentence arise at least once by random chance. What if we add mutation and natural selection (admittedly with high mutation rates and strong selection to emphasize the point)? If we run the simulation on this website, starting with 1024 random sequences, allow them to mutate and be subject to natural selection, it can achieve (in the simulation I just ran) the desired sentence in 257 generations and a total of 66,816 tries (which is a lot less than 10⁵² tries!).

The key to this process is that it's cumulative. As long as the fitness of intermediates between a completely random sequence and the target sequence have their fitnesses correlated with how close they are to the target sequence, then random mutation and non-random natural selection allow relatively rapid evolution of the target sequence. The vastly improbable becomes very probable.