Suppose that we have two hypotheses about a long sequence *X* of zeroes and/or ones:

*S*: The sequence has the underlying structure of a sequence of independent and identically distributed (iid) probabilistic phenomena with possible outcomes zero or one.*U*: The sequence is completely probabilistically unstructured, being completely probabilistically inscrutable.

What can we say about the evidential import of *X* on *S* and *U*?

Here is one thought. We have a picture of how a repeated sequence of
independent and identically distributed binary events should look. Much
as van Inwagen says in his discussion of the replay argument, we expect
to have the proportion of ones to oscillate but then converge to some
value. So if *X* looks like
that, then we have evidential support for *S*, and if *X* doesn’t look like that—maybe there
is wilder oscillation and no convergence—then we do not have evidential
support for *S*.

But does the fact that the sequence *X* has a “nice gradual convergence”
in frequencies support the structure hypothesis *S*? Van Inwagen in his replay
argument against indeterministic free will seems to assume so.

Actually, convergence of frequencies does not support *S* (or *U*, for that matter). Here is why
not. Let’s model hypothesis *S*
as follows. There is some unknown real dispositional probability *p* between 0 and 1, and
then our sequence *X* is taken
by observing a sequence of iid results with the probability of the
result being 1 being equal to *p*. Given *p*, any sequence of *n* events whose proportion of ones is
*m*/*n* has probability
*p*^{m}(1−*p*)^{n − m},
regardless of whether the sequence shows any “nice gradual convergence”
of the frequency of ones to *m*/*n* or is just a seqence of
*n* − *m* zeroes followed
by *m* ones.

Of course, hypothesis *S*
doesn’t say what the probability *p* of getting a one on a given
experiment is. So, we need to suppose some probability distribution
*Q* over the possible values of
*p* in the interval [0,1], and then our probability of the
sequence *X* of *m* ones and *n* − *m* zeroes will be ∫*p*^{m}(1−*p*)^{n − m}*d**Q*(*p*).
But it’s still true that this probability does not depend on the order
between the ones and zeroes. The sequence could “look random”, or it
could look as fishy as you like, and the probability of it on *S* would be exactly the same if the
number of ones is the same.

On the other hand, on the hypothesis *U*, *all* possible sequences
*X* of length *n* are probabilistically inscrutable,
and hence we cannot say anything about any sequence being more likely
than another—we might as well represent the probability of any sequence
*X* of observations as just an
interval-valued probability of [0,1].

So, no facts about the order of events in *X* make any difference between the
hypotheses *S* and *U*. In particular, van Inwagen’s idea
that an appearance of convergence is evidence for *S* is false.

Now, let’s say a little more about the Bayesian import of the
observation *X*. Let’s suppose
that *S* and *I* are our only two hypotheses, and
that both have serious prior probabilities, say somewhere between 0.1 and 0.9.
Further, let’s suppose that that the sequence *X* is long and has a decent amount of
variation in it—for instance, let’s suppose that it has at least 50
zeroes and at least 50 ones. Because of this, *P*(*X*|*S*) will be
some astronomically small positive number *α*. Indeed, we can prove that *α* is at most 1/2^{100} ≈ 10^{−30}, for any
distribution of the unknown probability *p* of getting a one.

On the other hand, *P*(*X*|*U*) will be
completely probabilistically inscrutable, and hence can be represented
as the full interval [0,1].

Here’s what follows from Bayes’s theorem combined with a natural way
of handling inscrutable probabilities as a range of probability
assignments: The posterior probability of *S* will be a range of probabilities
that starts with something within an order of magnitude of *α* and ends with 1. Hence, our observation of *X* does not support *S*: upon observing *X*, we move from *S* having some moderate probability
between 0.1 and 0.9, to a nearly completely inscrutable
probability in a range starting with something astronomically small and
ending at one. The upper end of the range is higher than what *S* started with but the lower end of
the range is lower.

Thus, if we were to actually do the experiment that van Inwagen
describes, and get the results that he thinks we would get (namely, a
sequence converging to some probability), that would not support the
hypothesis *S* that van Inwagen
thinks it would support.