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I am trying to define an AI domain in which I need to define the probability $p_i,$ of a variable $v_i$ having a value 1 or 0. $p_i$ may range anywhere from 0 to 100%. In my problem i goes from 1 to 83. $p_i$ has the following restriction:

I want on average for 47 of the variables $v_i$ to take value 1, so

$$\sum_{i=1}^{83} p_i=47.$$

And they need to have an extra characteristic due to my domain which is obeying this formula:

$$\sum_{i=1}^{82}\dfrac{\sum_{j=i+1}^{83}\frac{p_i(1-p_j)}{p_i(1-p_j)+(1-p_i)p_j}}{83\cdot82/2}=80\%,$$

Any thoughts on how do I solve this?

(For reference, here is the original link.)

EDIT (from comments):

I am trying to define an AI domain, those are probabilities (p_i) of several variables being 1 or 0. But I have to have more 1's on smaller i's and more 0's on larger ones. On average, I should have 47 1's, meaning that the sum of their probability should be 47.

I will actually be varying the values of the 80% and the 47, but I thought it would be easier to post it like that.

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    I texed your equations. Please check if I introduced any errors in doing so.2012-08-15
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    The first is not a linear equation, so the tag (linear-algebra) should be changed.2012-08-15
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    I changed the tag to (algebra-precalculus); if someone thinks of a more suitable tag feel free to change it.2012-08-15
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    Ok, Great, thank you three2012-08-15
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    Consider one specific i and j, being i2012-08-15
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    @Monica, please delete your comments and add the additional information into the question. I think others are more likely to help that way.2012-08-15
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    OK, but do you think I made myself clear on the last one? thank you for helping me ;)2012-08-15
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    I changed it!!!2012-08-15

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