The letter Q in Scrabble

How many you can expect

There is only one, so a rack either has it or it doesn’t — no middle case, and 93% of opening racks don’t.
Don’t read that as a per-game figure, though. Over a two-player game the bag empties, so you personally draw something close to half of it — which means you hold this tile in roughly half your games, not 7% of them. The 7% above is one draw: the opening rack.

Is it a bingo tile?

A rack picked at random can play all seven tiles 12.6% of the time. A rack holding at least one Q manages 1.5% — only 0.12× the baseline, which makes it the worst bingo tile in the bag. High face value and a hard rack tend to be the same thing. 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 reason is countable rather than folklore. 1.4% of the 19,424 seven-letter words in ENABLE use a Q somewhere in them. A tile that appears in more words is a tile more racks can build around, and that is the whole of the effect above.

One caveat worth stating: this is a fact about whole opening racks holding a Q, 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.

What it’s worth carrying

Q is worth 10 points, which puts it 1st highest-scoring tile of 27 — tied with 1 other at the same face value. Face value tracks rarity by design, but not as hard as people assume: the whole set is worth 187 points, and the five one-of-a-kind tiles (J, K, Q, X, Z) carry only 22% of that between them. The ten one-point letters carry 36% — more than the whole high-value end, because there are so many of them.