Evolution (the biological fact) is fuel by something that is the exact opposite of random: natural selection. Let me say that again, biological evolution is driven by something not random. Natural selection is the differential survival and reproduction of organisms in a population. I've repeated that to creationists so many times its not even funny. I often wonder if, because of their lack luster education and understanding of science, they have difficulty understanding the word differential. So let me say it in another way: the chance that one organism in a population will survive and reproduce is not the same chance that another member of the population will. If evolution were random, the chances that each organism in a population would survive and reproduce would be equal.
Considering this is something I've covered before I have a nice post on the math and 'chances' of evolution typed up for you already;
Please address it Alter2Ego
To search for a particular gene we are going to do as Dembski suggests and rely upon blind chance alone. The target in this case is, or sequence of our gene is 6/6/6 (that's a 6 on each die).
Throwing one die we have a 1/6 chance to land on a 6 and 5/6 chance to land on anything but 6. Thus we have 6 possible outcomes of our roll. With 2 dice we would have (6*6) 36 potential outcomes and with 3 dice we would have 216 potential outcomes (6*6*6).
Of all outcomes, only one is 6/6/6 so we have chance of 1/216. Certainly not astronomical odds, but considering our limitation of generations, I wouldn't bet on it.
So how many generations would we expect to go through before we reach our target gene?
Its going to get a little more mathy, but I think you'll be able to follow along. If not, just point out where I'm loosing you.
Let's, for arbitrary reasons, call the probability we succeed on the first roll, A. The probability then for succeeding on the second roll would simply be A(1-A). (Follow how we did that?)
So the probability of succeeding in finding the target on the third roll is simply A(1-A)(1-A). See the pattern evolving here?
The fourth roll; A(1-A)(1-A)(1-A) etc. We can rewrite this as A(1-A)3 =A4, where A4 simply denotes the roll.
From this we can derive a simple statistical rule. Since the term (1-A) is simply multiplied to the rule for each additional roll, we can say that the probability to succeed on any roll (An) is simply A(1-A)(n-1).
For example, the probability to find our target gene on the 27th roll (generation) is simply A(1-A)26.
Since, the game must be completed by a certain generation, we have an expectation for completion which we'll arbitrarily call E.
The expectation is simply the sum of all of the probabilities of each round till the game is won. Mathematically that simply means that;
Since our expectation has to be a positive real number, we know it can be any number from 0 to infinity. Taking the limit of the above equation we would get;
A*1/A2=E, where is the expected number of generations to find our target.
To solve this, remember that A is the probability of succeeding on the first roll or 1/216. So plugging that into our equation we get E=216.
Obviously then, we wouldn't expect evolution to produce target complexity without the intervention of an intelligent agent. Or would we?
In this game we are going to play more akin to how I described evolution above. By using blind trials but saving positive outcomes into the next generation. A positive outcome in this example would be a 6.
So whenever a die is rolled that lands on a 6, that die is saved to the next round (a free pass).
We can then go about calculating the number of rounds or generations we would expect to play to win the game.
Consider when we roll 1 die, we had a 1/6 chance of 6 and a 5/6 chance of not 6. To calculate our expectation, arbitrarily defined as x, we;
x1 = 1 + 5/6 * x1
1/6*x1 = 1
x1 = 6
With 2 dice we have a 1/36 chance of finishing the game in one step 6/6, a 10/36 chance of rolling one 6 and a 25/36 chance of rolling no 6 at all, thus;
x2 = 1 + 10/36* x1 + 25/36* x2
11/36*x2 = 1 + 10/36 * 6 (the 6 comes from the above answer to equation 1)
x2= 36/11 * 8/3 = 96/11 or that is we expect on average to play the game 8.72 rounds
So with 3 dice we have a 1/216 chance of finishing the game in one step, a 75/216 chance of rolling a single 6, a 15/216 chance of rolling two 6’s and a 125/216 chance of rolling no 6 at all.
So our equation becomes;
x3 = 1 + 75/216 * x1 + 15/216 * x2 + 125/216 *x3
91/216*x3 = 1 + 75/216 * 6 + 15/216 * 96/11
x3 = 216/91 * 487/132
x3 = 8766/1001 or 8.76 rounds
We then would most often, win the game. The implication of this then, if you haven't followed it through the two games, is that by adding selection and heredity into the mix evolution by natural selection is capable of generating complex information. And in it does this in a evolutionary timely manner with the assistance of heredity and selection.
In fact, the argument gets worse for Dembski, the more we liken it to the real world. Because in evolution, there are many, many, many simultaneous trials (organisms) each playing the game.