Originally posted by dj2becker
Suppose we put chance to a test which is less simple, yet something that would be quite easy for any school child. Let it spell this phrase: “the theory of evolution.” Drawing from a set of twenty-six small letters and one blank for the space between letters, what is the probability expectance?
All that is needed is simply to get those twenty-three ...[text shortened]... 0,000,000 times the assumed age of the earth!
A couple of problems. First, as LJ, already pointed out. The analogy is a poor one to evolution.
1) It assumes that organisms appear in there current state. In the post, attempts at spelling "the theory of evolution" are pulled in one 23 long string. A better analogy would have the machine draw letter by letter and build on with a mechanism that rejected strings that were not going to create "the theory of evolution" just as natural selection weeds out certain combinations. This would greatly reduce the amount of time needed for an a priori 50% chance of success. A lot of cell phones actually use something like this for text messaging. On my phone it is called 'predictive text'.
2) It assumes that life sprang from non-life in the manner in which we find it today. That is it assumes evolution never actually occured. In the analogy "the theory of evolution" is the only acceptable response. We do know life as it appears on earth today is not the only possible way in which it can form. The analogy then should allow for more phrases (including shorter ones) to also qualify as a success.
3) By the logic in the post, in a letter-by-letter draw, the probability of drawing any subsequent letter would be the same given what has been drawn before. This certainly is not true with evolution. New organisms evolve gradually and remain fairly similar for a long time.
Second, the math is wrong (who'd have thunk it!). The calculation of 27^23 is correct, so is the calculation of the number of draws per year. The error is in saying that it will take so many years for the phrase to be spelled. Ex ante the number of trials needed to have a 50% chance of success is much lower.
I cannot calculate it exactly with their example because the exponents are too large to calculate easily. I can, demonstrate the principle though if you will permit me to design an analogous example. Then I will show that this applies to your example and even ones with arbitrarily large numbers.
Let's say we want to spell "toe." The total number of possible draws is 26^3 or 17576. So drawing at random the probability of drawing "toe" on the first try is 1 in 17576. Now using the logic in your example, if we drew once every year, we would need 17576 draws or 17576 years to have a reasonable chance of getting "toe". This is false. Allow me to show my calculation.
What is the ex ante probability of getting "toe" in 17576 or fewer attempts?
Ans: 1-(the probability of not getting "toe" in 17576 tries)
1-((17575/17576)*(17575/17576)*. . .)
= .6321 approx.
Notice that the second term is ((n-1)/n)^n. In the limit as n goes to infinity this goes to e^-1 which equals approximately .367879. So even for your example with very large numbers the probability of spelling "the theory of evolution" in the number of possible attempts in 26 billion years or whatever is a bit more than 63%.
Back to my example, how many years would we need to allow ex ante to have a 50% chance of drawing "toe"?
x = ln (.5)/ ln(17575/17576)
= 12182.4 years
30% less time than stated by the logic in your example.