First off, when searching for this question I didn't find anything exactly like what I'm asking, but if someone finds a duplicate post please let me know.
Here is my question:
Say that you have an infinite amount of sets, each containing one blue object and ten green objects, totaling 11 objects in each set.
Then say we have an infinite set P as the set containing all these little sets.
What are the probabilities that, reaching into set P, you'd pick out a blue or green object? Are they still 1/11 and 10/11, respectively?
Thanks.
Let's call $S_i$ each small set, with $i=1, 2, \ldots, +\infty$. Suppose that $p_i$ is the probability to pick the set $S_i$. Then, you know that:
$$P(\text{blue} | S_i ~\text{has been picked}) = \frac{1}{11},$$ $$P(\text{green} | S_i ~\text{has been picked}) = \frac{10}{11}.$$
Moreover:
$$P(\text{blue}) = \sum_{i=1}^{+\infty}P(\text{blue} | S_i ~\text{has been picked})p_i = \frac{1}{11}\sum_{i=1}^{+\infty}p_i,$$ $$P(\text{green}) = \sum_{i=1}^{+\infty}P(\text{green} | S_i ~\text{has been picked})p_i = \frac{10}{11}\sum_{i=1}^{+\infty}p_i.$$
By definition,
$$\sum_{i=1}^{+\infty}p_i = 1,$$
then
$$P(\text{blue}) = \frac{1}{11},$$ $$P(\text{green}) = \frac{10}{11}.$$