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 categorical distribution is the most general and has parameters to assign a probability to every single outcome.
$$\begin{array}{rcl}
P(y = 1) &=& p_1\\
P(y = 2) &=& p_2\\
P(y = 3) &=& p_3\\
P(y = 4) &=& p_4\\
P(y = 5) &=& p_5\\
P(y = 6) &=& p_6\\
P(y = 7) &=& p_7\\
\end{array}$$
When you are looking for something like a geometric distribution or other distribution with less parameters, then there is some continuous mechanism, like a coin flip for each turn, and you can use the properties of that coin flip to parameterize the distribution.
As Scortchi mentions in the comments, such a mechanism is not necessary in order to use a parametric distribution. But these parametric distributions tend to relate to some mechanism and when you apply the parametric distribution then you could see the related mechanism as a model for the underlying mechanism of the distribution.
For the wordle outcomes it is unlikely that you can generate a distribution based on a single parameter or few parameters. There are too many factors and effects in order for the distribution to be easily modelled by a few parameters.
You have many mechanisms that are not much related

*

*The probability to make the first guess


*The probability to make the second and third guess based on how the first few hints worked out.
(In my personal strategy these probabilities are zero. Even if I would have only one letter left over to guess, then I would still try a different wordt that incorporates multiple letters)


*The probability to make it in the fourth or fifth guess with more hints. Depending on how well the initial words match one can guess in the fourth or fifth. But I don't think that these can be modelled with only a single parameter.


*The probability to not guess after six.
If the algorithm to guess the word fails. With a good strategy the probability should be nearly zero. This probability requires a seperate parameter and is unrelated to the probabilities to guess in 4 or 5 turns.
These mechanisms are much unrelated and can not be easily combined and modelled by a smaller set of parameters.
It is also difficult because guessing words leads to an algorithm that is not very uniform. It depends on what you get. If you would instead have something like mastermind then you could create something like a parametric distribution (e.g. the probability to guess in turn $k$ as function of the length and number of colours).
