which resembles a binary
number when written without punctuation
There are several every century: the first, tenth and eleventh of January, October and November of 00, 01, 10, 11. That's 36 a century: less than 0.1% of all days are binary days.
This node brought to you by The League for Useless Reasons to Celebrate. And I'm going down the pub tonight for just this reason! (Well, and the pub quiz).