Probability of hitting X shots in N tries knowing that the P(hit) is the ratio of previous hits Let $X$ be the number of hits in $N$ tries, I know that the probability of the next hit is $P(\text{Hit}) =X/N$.
How can I get the generic expression for the probability distribution function of hitting $x$ hits in $N$ tries, knowing that the probability of hitting the first shot is $p_1$?
I've been trying to find the solution to this, I know from brute forcing that for $p_1 = 0.5$ the distribution is a constant ( $1/(N+1)$ I believe), when it's higher than 0.5 it's a positive slope line and lower than 0.5 a negative slope line. I just don't know how to mathematically reach a result.
Edit: Clarification.
Let's assume the first two 6 shots are known. There were 3 hits and 3 misses, and so the probability of the 7th shot hitting is 0.5. Should that hit, the probability of hitting the 8th shot would be $4/7$. Should that miss, the probability of hitting the 8th shot would be $3/7$.
I forgot to say that when I was brute forcing I assumed the first two shots to be a hit and a miss, but knowing any sequence of two or more previous shots would be necessary for the problem to make sense.

"There were 3 hits and 3 misses, and so the probability of the 7th shot hitting is 0.5. " Do you mean: "There were 3 hits and 3 misses, and so my estimate of the probability of the 7th shot hitting is 0.5. "
Some more information is needed before your question can be answered. Firstly, is the process stable? i.e. is the real probability of a hit constant, or does it drift over time?
Secondly, what prior knowledge do you have of the process? If you can encode your prior knowledge as a probability distribution of the unknown P then bayesian methods provide a precise answer to your question.
In the absence of prior knowledge your question has no answer. Put it this way: suppose I have already planned the infinite sequence miss, miss, hit, hit, miss, hit, miss ... according to some whim, then the first $n$ results don't really tell you anything about the $n+1^{st}$
