I'm a little lost regarding the concept of estimators of probability in regard to the two following examples.
Suppose I have $n$ marbles in a bag, and I sample them many many times and find that $20$ percent of the times, a red marble shows up. Since I don't know the actual number of red marbles in the bag, I've been told this $P=0.2$ is the estimator of the true probability, can someone verify this for me ?
Moving on, now I want to make a distribution of the number of red marbles in the bag. I take $n+1$ copies of the bag, each with a fixed number of red marbles, and I check which of these is most likely to be our original bag. The original bag had a $20$ percent chance of giving a red marble, so I could expect to find $20$ red marbles out of $100$. Now I sample, each of these bags, and check which one gives me $20$ marbles out of $100$ the most times. For example, the bag with all red marbles, will never give me $20$ red marbles out of $100$ ( it'll always give me a red marble ) and so, it is definitely not our original bag.
This distribution would be modeled by: $$ p= {^{100}C_{20}} \left(\frac{n_i}{n}\right)^{20} \left(1-\frac{n_i}{n}\right)^{100-20} $$
Here, $\frac{n_i}{n}$ is the probability of drawing a red marble from each of these bags.
This distribution tells us, what is the probability of a bag with $n_i$ red marbles out of $n$, of being our original bag, which gave me a $20$ percent chance of drawing a red marble.
My problem is, this is slightly different from the binomial distribution that I'm used to seeing. Here, the number of successes is constant, while the probability $\frac{n_i}{n}$ is variable. In a sense, we are calculating the probability of a bag of giving us a certain probability of drawing a red marble - a probability of probabilities.
Can anyone tell me what this is, and what an estimator is, in regard to this particular example ?
In a different answer on a question, I read the following :
A distribution for a random variable says given the parameter m, what is the probability that the data comes out to x. The corresponding estimator says given the data came out to x, what is the probability that the parameter was m.
So, in my example, suppose the number of red balls in a bag was $20$ out of $100$. A probability distribution would tell me, the probability of picking up $100$ balls with replacement, and finding that $20$ are red.
On the other hand, if I picked up a $100$ balls, and found that $20$ were red, an estimator would tell us the probability that there were initially $20$ red balls in the bag, or rather, the probability of drawing a single red ball was $0.2$ originally. In this sense, the estimator is sort of a probability of probability.
Is this analogy correct ?