Composing dice throw probabilities Suppose we are given a series of probabilities $p_a=0.2, p_b=0.1, p_c=0.5$ and $p_d=0.3$, for obtaining the value $4$ in a fair-dice throw. But the estimates were obtained for varying number of throws, e.g. the estimate of experiment $a$ was obtained after $5$ throws, so $n_a=5$, for others $n_b=10, n_c=3, n_d=6$. 


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*Question is, how should one correctly compose these probabilities, in order to give a final estimate $p$ from the gatherrd statistics? Should one be careful of the normalization used for the $p_i$'s? 


My first attempt was to take a weighted average of them, using the ratio between individual throws and total throws as weight, i.e.:
$$n=\sum n_i$$
so
$$p=\frac{n_a}{n}p_a+\frac{n_b}{n}p_b+\frac{n_c}{n}p_c+\frac{n_d}{n}p_d$$


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*Is this a valid way of going about composing the probabilities of different experiments?

 A: As many interesting points have been already pointed out in the comments, and they in essence already contain the answer to the post, I decided to write them up here as an answer for future readers.

Mainly extracted from lulu's comments:
Since the individual probabilities in each experiment were obtained by dividing the total number of occurrences of value $4$ by the total number of throws of that experiment, the weighted average given for $p$ in the question, boils down to simply counting the overall occurrences of $4$'s from all experiments and dividing by the total number of throws $n.$ Thus one is not really composing probabilities, rather just doing the right counting. 
Moreover as we were only interested in observing the value $4,$ we essentially treat all other values as failures, which means we can use the binomial distribution as model of choice. Otherwise the multinomial distribution can be used in a more general context.
Finally as to alternative ways of estimating the probabilities, lulu nicely points out:

there are other sensible ways of estimating the probabilities. You
  could, for example, just start by assuming that $p=1/6$ then
  re-estimate using Bayes' Theorem as you get new data. In that way,
  throwing a $4$ first wouldn't change your mind much, but starting out
  with five $4$'s in a row certainly would. Doing it this way would make
  your "composing" problem a lot more delicate...as you'd need to keep
  track of the individual orders (and declare what you meant by the
  composite order).


Important points brought up by joriki, as the post admittedly lacks in rigour:
From the scheme given in the post for computing the probabilities $p_i,$ it follows logically that the probabilities should be multiples of $1/n_i,$ $n_i$ being the number of throws in experiment $i.$ But the chosen numerical values in the post do not seem to fulfill this condition, as a result of a blunder in arbitrarily choosing the values. 
This point is of important relevance, as joriki points it out:

The discussion was based on the premise that these were estimates
  obtained by dividing the number of 4s by the total number of throws in
  each case, and then your weighted calculation reconstructs the optimal
  estimate for the combined experiment. But if this is not how the
  estimates were obtained, we can't say much about how they should be
  combined.

