If you pick a set $X\subset\{1,2,...,n\}$ by including each number $i$ in $X$ with probability $p_i$ at random, then you pick an element $x$ from $X$ uniformly at random. Is it true that $\mathbb{P}(x=a)=\frac{p_a}{\mathbb{E}[|X|]}$?
Picking a uniformly at random element from a random set
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probability
probability-distributions
2 Answers
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Take the following example: $\{1,2\}$. Let the probabilities of picking $1$ or $2$ be $p_1$, $p_2$ respectively.
The expected size of the subset we select is
$$E=p_1(1-p_2)+p_2(1-p_1)+2p_1p_2=1.$$
The probability that we pick $1$ from the randomly selected subset is
$$P(1)=P(1\mid\{1\})p_1(1-p_2)+P(1\mid\{1,2\})p_1p_2=p_1(1-p_2)+\frac12p_2p_2.$$
So $$P(1)\not=\frac{p_1}E.$$
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What I can tell you is that $$\mathbb{P}(x=a)=p_a\sum_{i\in[n-1]}\sum_{Y\in\binom{[n]\setminus \{a\}}{i}}\left(\prod_{z\in Y}p_z\right)\left(\prod_{z\notin Y}(1-p_z)\right)$$ It can or not be the expression you said