What statistical distribution would best capture a set of Wordle outcomes? Wordle is a simple and popular word game. It is based on an older game, and has recently gone viral and attracted attention. It is available in several languages and the original site (now owned by The New York Times is here)
The rules are simple. Each day, there is a secret target five letter word. You guess a five letter word, and you get hints of which letters in your guess belong to the target word and whether the letter is in the same position. You have six tries after which the letter is revealed. That is, the outcome is the number of tries, which we can index by $1, ..., 6$ and assign the $7$ to the failure to guess the word.
Let's say I play Wordle 100 times and I want to model the distribution of outcomes. Each game yields an outcome $y \in \{1,...,7\}$. In order to simplify the question, I am assuming that failing and succeeding in seven tries is the same. I am not so much interest in Wordl but on the statistical distribution of this outcome.
If we were just flipping a coin until seeing a Head, we would have a geometric distribution. But in Wordle each try gives hints and changes the probability: it is close to impossible to guess the word in one try and pretty common to guess the word in 4 to 6 tries, and the geometric distribution does not capture this.
How would you model this distribution? It seems like "cheating" to try to fit something more flexible like a Negative Binomial just to fit the data. Is there a distribution that generalizes the geometric distribution with a probability parameter that changes monotonically after ever try?
Just to clarify, I don't care about linking the peculiar characteristics of World itself. I Just want something more flexible than the geometric distribution, without having to go full non-parametric and model the whole thing using seven parameters.

The probability models being suggested seem to depend on considering the selection of letters as equally likely. This does not seem very realistic to me. The word choice for #1 comes from the 26 characters in the alphabet, letters 1, 2, 3, 4, 5.
Vowels a,e,i,o,u and y have an order from most to least likely and consonants have a similar order. This to me would be the starting point.
