I don't think anyone has pointed you at quite the right direction for getting a fully satisfactory answer to your question. I think what you're looking for is the concept of Continuous Universal A Priori Probability:

The universal distribution m is defined in a discrete domain, its arguments are finite binary strings. For applications such as the prediction of growing sequences it is necessary to define a similar distribution on infinite binary sequences. This leads to the universal semi-measure M defined as the probability that the output of a monotone universal Turing machine U starts with x when provided with fair coin flips on the input tape.

For more details see the linked article, or if you're really interested in this field, get the referenced textbook by Li and Vitanyi.

EDIT: On second thought I'll spell out what I think is the answer, instead of just giving you this hint. This form of Solomonoff Induction, when faced with a growing sequence of fair coin flips, will quickly assign high probability to input tapes that start with the equivalent of "copy the rest of the input tape to the output tape as is", and therefore can be interpreted as assigning high probability to the hypothesis that it is facing a growing sequence of fair coin flips.