The blank tile
2 in the bag · 2% of all tiles
Drawing 7 tiles from a full bag, the blank tile shows up about one rack in 7 — the 13th likeliest tile to see of 27 (tied with 9 others, since every tile with the same number of copies has exactly the same chance).
How many you can expect
Most racks hold none at all — 86.4% of them, and 0.42% hold two or more. On average a rack carries 0.14 of them.
Exact chance of holding each number of Blank tiles in an opening rack of 7.
Is it a bingo tile?
A rack picked at random can play all seven tiles 12.6% of the time. A rack holding a blank manages 45.0% — 3.56× the baseline, and the best bingo tile in the bag. Measured against ENABLE, which is public domain. The official lists are both bigger, so both would give a higher number — a little higher for NWL, appreciably so for Collins.
The blank is the extreme case of the same thing: it is not in any word, so it is in all of them. Whatever letter the rack is short of, it becomes — which is why the two tiles that score nothing are the two most valuable in the bag.
One caveat worth stating: this is a fact about whole opening racks holding a blank, averaged over every set of six tiles it could arrive with. It is not a leave value. Whether to keep this tile mid-game depends on the six tiles beside it and on what is left in the bag, and a tile that is good on average can still be the wrong keep.
Why the blank is worth more than it scores
A blank scores nothing, which is why it drags the average rack value down rather than up. It is still the tile most players would rather draw than any other: it is the only one that can be whatever the board needs. There are just 2 of them in 100 tiles, so the odds above are the whole story — nothing about a blank varies by letter.