In Specification: The Pattern that Signifies Intelligence, William Dembski revisits the idea of a "universal probability bound" (see also "The Design Inference"). This is a useful concept - as a bound, it is "impervious [to] any probabilistic resources that might be brought to bear against it." If the probability of a specified event is smaller than this, then it basically won't happen by chance.
The word "specified" is important. Consider the probability bound of 10-150 that Dembski suggests. A computer could generate a random sequence of 100 letters, spaces, commas and full stops. Each 100 character sequence has a probability of one in 10-150 of arising. However, the program has to produce something - an improbable event in itself isn't significant. But suppose I wrote down a particular series of 100 letters, spaces, commas and full stops (with no knowledge of the algorithm that was used by the program). What is the likelihood of the program generating this sequence that I have specified? It is (roughly) 1 in 10150 - and with this universal probability bound, I am able to assert that this is unlikely to arise by chance in the life of the universe - and by extension, in however long the program runs for. So if the program generates this specified sequence, then somebody has cheated somewhere. Or, in Dembski's terms, we can draw an inference of design.
This figure of 10-150 is based on "the number of state changes in elementary particles throughout the history of the universe." It is reasonable to regard this as a "worst possible case" - the universe has a maximum total number of 10150 goes at anything that can be possibly conceived of - so if the probability of a specified event happening by chance is less than the reciprocal of this, we would conclude that it didn't happen by chance.
If this figure is the universal probability bound, I'd like to propose the introduction of slightly different but related probability bounds - perhaps we could call them biochemical, biological, thermodynamic and chemical probability bounds. Considering biological systems, for example, the total number of state changes in the universe is not really relevant. Fast biochemical reactions are fast, no doubt, but whereas the Planck time used to derive this universal probability bound is 10-45 seconds, the fastest biochemical reactions require a period of time at least some 30 orders of magnitude greater (the initial stages of photosynthesis reactions take a period of time of the order of 10-12 seconds. We might allow that most of the history of the universe is relevant for consideration of biochemical processes - but since the majority of the matter of the universe is hydrogen and helium (all but one thousandth of 1 percent), given that biochemical reactions basically involve mass of the universe that isn't dominated by hydrogen and helium, it would also seem to be sensible to reduce the biological probability bound by five further orders of magnitude from the cosmic probability bound. Thus an estimate for the biochemical probability bound would be 10-115.
It seems odd to be fiddling around with orders of magnitude when we are discussing such small numbers. However, probability bounds have relevance to consideration of real scientific processes, so it helps for such discussions to be based on realistic estimates. I intend to explore the implications of the biochemical probability bound in a subsequent post.