Uniquely Genius Queen Bees

In Spelling Bee, a word game from The New York Times, you must create words using seven letters arranged in a honeycomb grid. Letters can be used more than once per word, but each word must be at least four letters long and include the central letter. You may recall a previous puzzle (RIP, FiveThirtyEight) based on Spelling Bee.

Each day, your goal is to score as many points as possible. Four-letter words are worth 1 point, longer words are worth the number of letters they contain, and “pangrams” (words containing every letter) provide 7 bonus points. But these specific details don’t matter for this week’s puzzle.

If you find all the words in a given day and therefore accrue the maximum number of points, you earn the “Queen Bee” ranking. Meanwhile, accruing smaller point totals earns you other rankings. In particular, the point cutoff for “Genius” is 70 percent of the maximum number of points, rounded to the nearest whole number. While the cutoff for “Genius” is clearly displayed in the puzzle, the “Queen Bee” point total is not readily shown.

Of course, this maximum total can be approximated by dividing the “Genius” cutoff by 0.7. Even so, the total may be ambiguous, since multiple “Queen Bee” values can result in the same “Genius” cutoff.

Suppose a given round of Spelling Bee has some very large, randomly chosen point total. What is the probability that this total can be precisely determined (i.e., without any ambiguity) from its point cutoff for “Genius”?

Let's assume that $Q$ is the Queen Bee point total and that $G$ is the Genius cutoff, and we have the functional definition $G: \mathbb{N} \to \mathbb{N}$ with $Q \mapsto G(Q) = [ 0.7 Q ],$ where $[\cdot]$ denotes the closest integer function, that is $[t] = \min \arg\!\min \{ |n - t | \mid n \in \mathbb{N} \},$ where we are rounding halves downward towards $0$. While the concept of uniform distribution on the natural numbers is untenable, let us assume that here we mean by some very large, randomly chosen point total of $Q$ that we have the following probabilistic definition. For any subset $A \subseteq \mathbb{N}$, let us define its probability as $$\mathbb{P} (A) = \lim_{n \to \infty} \frac{ \left| A \cap \{1, 2, \dots, n \} \right| }{n}.$$ This is loosely a limit of the sequence of uniform distributions on the sets $\{1, 2, \dots, n \}$ as $n \to \infty,$ but again I don't really want to get into the formal definitions of how that limit should be defined, let's just kinda run with it.

Anywhoozle, in this case, let's define $U = \{ n \in \mathbb{N} \mid |G^{-1}(n)| = 1 \}$ to be the desired set of all values of $Q$ that can be precisely and unambiguously determiend by the corresponding value of the $G$ function. It would probably be good to have some idea of what $U$ looks like before plowing ahead and getting to the calculation of the desired answer, that is, $\mathbb{P}(U).$ Let's start super simple and just give the first few values of the $G$ function:

q	0.7q	G(q)
1	0.7	1
2	1.4	1
3	2.1	2
4	2.8	3
5	3.5	3
6	4.2	4
7	4.9	5
8	5.6	6
9	6.3	6
10	7.0	7
11	7.7	8

Using these first 10 integers, we see that $3, 6, 7, 10 \in U.$ We also can see that for any $d \in \mathbb{N}$ that $$G(10d+k) = \left[0.7 \left(10d+k\right)\right] = 7d+G(k).$$ So we can surmise that in face for any $Q \in \mathbb{N}$ if $Q\! \mod\! 10 \in \{0, 3, 6, 7 \}$ then $Q \in U,$ then $$U = \left(10 \mathbb{N} \cup (3 + 10 \mathbb{N}) \cup (6 + 10\mathbb{N}) \cup (7 + 10\mathbb{N}) \right).$$ We see that $$\left| U \cap \{ 1,2, \dots, n\} \right| = \begin{cases} 4 \lfloor \frac{n}{10} \rfloor, &\text{if $n \!\mod\! 10 \in \{0,1,2\};$}\\ 4\lfloor \frac{n}{10} \rfloor+1, &\text{if $n \!\mod\!10 \in \{3,4,5\};$}\\ 4\lfloor \frac{n}{10} \rfloor+2, &\text{if $n \!\mod\!10 \equiv 6;$}\\ 4\lfloor \frac{n}{10} \rfloor+3, &\text{if $n \!\mod\!10 \in \{7,8,9\}.$}\end{cases}$$ Using the squeeze theorem and the fact that $$\lim_{n\to \infty} \frac{4 \lfloor \frac{n}{10} \rfloor}{n} \leq \mathbb{P} (U) \leq \lim_{n\to \infty} \frac{4 \lfloor \frac{n}{10} \rfloor + 3}{n},$$ we have the probability of getting a value of $Q$ that is precisely, unambiguously determined from its value of $G$ is $$\mathbb{P} (U) = \lim_{n \to \infty} \frac{1}{n} \left| U \cap \{1, 2, \dots, n \} \right| = \frac{2}{5} = 40\%.$$