The Design Inference by William A. Dembski
Cambridge University Press, 1998
p.xi, paragraph 1: "Just about everything that happens is highly improbable". He quotes several examples, one of which concerns an apparently randomly strewn arrangement of stones in a garden. "The precise arrangement of stones is vastly improbable".
I disagree that "just about everything that happens is highly improbable". I think that this is a misuse of the word improbable. I suggest that a correct expression of what the author had in mind would run as follows:
It is a common, everyday, experience that many different rare events happen.
The particular observed arrangement of stones is rare.
The words rare and improbable are not synonymous. A rare event is one that seldom happens. The term improbable (which is synonymous with unlikely) carries more baggage. Dembski does not define probability until p.70; his definition there is circular because it uses the undefined term likely which, in the relevant one of several meanings, is synonymous with probable, according to the dictionary.
The Concise Oxford English Dictionary (3rd edition, 1934, Clarendon Press, Oxford) defines probable as "That may be expected to happen or prove true, likely…". Webster's New Collegiate Dictionary (G. & C. Merriam Company, Springfield, MA, 1975) gives these definitions of probable:
Supported by evidence strong enough to establish presumption but not proof.
Likely to be or become true or real.
It seems to me that the following 3 items help one to use the term probable correctly:
There is a trial which (usually) could have more than one possible outcome.
There is an assessor (real or imagined) who, in ignorance (either real or assumed) of the outcome,
makes an assessment of his expectation about a particular possible outcome. If his expectation is strong, he would say that that outcome is probable.
Dembski's garden stones example involves only item a., namely, an act of strewing stones in the garden. Improbability does not enter into it.
On p.3, Dembski gives another example: "..plenty of highly improbable events happen by chance all the time. The precise sequence of heads and tails in a long sequence of coin tosses…".
Again, only item a. is present here, namely, the precise sequence of heads and tails in a long sequence of coin tosses. Any such sequence may properly be described as rare. Improbable would be a misnomer.
We can, however, construct a more complex, compound, hypothetical, trial or event, related to this example, whose outcome (if it actually occurred) may properly be described as being highly improbable:
a. The trial:
An experimenter first writes down a predicted outcome for a long sequence of N coin tosses to be made with a fair (unweighted) coin.
The experimenter then performs a sequence of N coin tosses.
The latter sequence agrees completely with the predicted sequence.
b. There is an assessor who, before part 2, makes
c. an assessment of his expectation about the outcome 3 as having a chance of success equal to (0.5)^N = 8 ´ 10^(-31) if
N = 100.
This event (i.e., 1. + 2. + 3.) is highly improbable.
Considering the above two trials involving coin toss sequences, we see that, in this case, the rare event happens once and only once and that the highly improbable event never happens. This statement could hardly differ more from Dembski's claim that improbable events happen all the time. One can produce a rare event (once) at any time one chooses. One cannot hope to produce a highly improbable event even once in a lifetime.