I have a difficult problem about probability that needs your help. Let $(x_1,x_2,x_3,x_4)$ be four unknown variables. From these variables, we can created a set of equations as follows:

$$ \begin{pmatrix} x_1 & & & \\ & x_2 & & \\ & & x_3 & \\ & & & x_4\\ x_1 +x_2& & & \\ x_1+ x_3& & & \\ x_1+ x_4& & & \\ & x_2 +x_3 & & \\ \vdots\\ x_1 +x_2+x_3& & & \\ x_1 +x_2+x_4& & & \\ &x_2 +x_3+x_4 & & \\ \vdots\\ x_1 +x_2+x_3+x_4& & & \\ \end{pmatrix} = \begin{pmatrix} d_1\\ d_2\\ d_3\\ d_4\\ d_5\\ d_6\\ d_7\\ d_8\\ \vdots\\ d_9\\ d_{10}\\ d_{11}\\ \vdots\\ d_{12}\\ \end{pmatrix} $$

The left side of the equation is created by four groups: the first group is created from a single variable, the second groups is a combination of two variables, the third groups is a combination of three variables and so on. The right side is a known vector $d$.

In the first phase, I will randomly pick $n_1$ equations from group 1, without replacement (each equation is chosen one time). The second phase, I randomly pick $n_2$ equations with replacement (means some equation can chosen more than one time) from all groups: 1 to 4, the probability of choosing group follows the rule:


These $n_1+n_2$ selected equation give us a new linear equation. My question is how to find the expected solved variables can we have from the $n_1+n_2$ linear equation?


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