I have two equations:
$$P_a=P_a(x,y)$$
$$ P_b=1-P_a(x,y) $$ $P_a$, $P_b$, $x$ and $y$ are probabilities so their values are between 0 and 1.
If I know the values of $x$ and $y$ I can calculate easily $P_a$ and $P_b$, the problems is I have $P_a$ and $P_b$, and I need to calculate $x$ and $y$. I had some ideas but any of them worked:
- The polynomials are high order and I tried to simplify the expressions, but I still get errors.
- I tried reduce also with assumptions, I still get the errors.
- I thought maybe the Inverse function could help but I have two equations and two variables and failed to use it in this case.
- I generated a list of points, then used Interpolation and the expression I got in the 3D Plot is similar to the original polynomial BUT I don't have any explicit expression to work with so it was not useful. Apparently there is an internal calculation inside Mathematica but you don't get an expression. I tried
InterpolatingPolynomialbut got an error. - I tried a Taylor expansion to get a simpler expression but only worked around $x$,$y\approx 0.5$. I saw in the plot that for values close to 1 there were some strange things.
So folks, does anybody know how could I get $x$ and $y$ as function of $P_a$ and $P_b$?
Or at least, does anybody know how to calculate $x$ and $y$ if I know $P_a$ and $P_b$?
I have lost 6 days with this problem and I am desperate.
This is the expression for $P_a$:
Pa = 1/(-wb + wa (-1 + 2 wb))* wa (-1 + wb) (-(-1 + wb)^5 (1 + 5 wb) + 15 wa (-1 + wb)^4 wb (1 + 5 wb) - 5 wa^2 (-1 + wb)^3 wb (-11 + wb + 70 wb^2) + 5 wa^3 (-1 + wb)^2 wb (17 - 69 wb - 28 wb^2 + 140 wb^3) - 5 wa^6 wb (1 - 14 wb + 56 wb^2 - 84 wb^3 + 42 wb^4) + 3 wa^4 wb (-23 + 202 wb - 473 wb^2 + 252 wb^3 + 252 wb^4 - 210 wb^5) + wa^5 wb (29 - 331 wb + 1064 wb^2 - 1176 wb^3 + 210 wb^4 + 210 wb^5)) I have to mention that I have the same problem with an expression that is like a 100 times larger than this one. Hopefully what I learn from this calculation could be applied in the more complicated polynomial.
I have to mention that sometimes when I push Enter to start a calculation Mathematica doesn't respond. Should I increase the memory dedicated to calculations to avoid this problem?
It says "running" but I can wait hours and I don't get an answer.
This is for example the code I used to try to Solve this polynomial:
NSolve[0.5 == 1/(-wb + wa (-1 + 2 wb)) * wa (-1 + wb) (-(-1 + wb)^5 (1 + 5 wb) + 15 wa (-1 + wb)^4 wb (1 + 5 wb) - 5 wa^2 (-1 + wb)^3 wb (-11 + wb + 70 wb^2) + 5 wa^3 (-1 + wb)^2 wb (17 - 69 wb - 28 wb^2 + 140 wb^3) - 5 wa^6 wb (1 - 14 wb + 56 wb^2 - 84 wb^3 + 42 wb^4) + 3 wa^4 wb (-23 + 202 wb - 473 wb^2 + 252 wb^3 + 252 wb^4 - 210 wb^5) + wa^5 wb (29 - 331 wb + 1064 wb^2 - 1176 wb^3 + 210 wb^4 + 210 wb^5)) && 0.5 == 1 - (1/(-wb + wa (-1 + 2 wb)) * wa (-1 + wb) (-(-1 + wb)^5 (1 + 5 wb) + 15 wa (-1 + wb)^4 wb (1 + 5 wb) - 5 wa^2 (-1 + wb)^3 wb (-11 + wb + 70 wb^2) + 5 wa^3 (-1 + wb)^2 wb (17 - 69 wb - 28 wb^2 + 140 wb^3) - 5 wa^6 wb (1 - 14 wb + 56 wb^2 - 84 wb^3 + 42 wb^4) + 3 wa^4 wb (-23 + 202 wb - 473 wb^2 + 252 wb^3 + 252 wb^4 - 210 wb^5) + wa^5 wb (29 - 331 wb + 1064 wb^2 - 1176 wb^3 + 210 wb^4 + 210 wb^5))) && 0 < wa < 1 && 0 < wb < 1, {wa, wb}] These are the errors I get:
NSolve::svars: Equations may not give solutions for all "solve" variables. >>
NSolve::ratnz: NSolve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. >>
And this is the result:
(*{{wb -> ConditionalExpression[wa, 0 < wa < 1.]}}*) I know the polynomials contain a lot of terms and could be high order, but the range is between 0 and 1, maybe there is a way to take advantage of that to get the solution.
ContourPloton both equations, you'll see that for both the solution is $wa=wb$ for $0 < wa, wb < 1$. So there are an infinite number of solutions. The result you see {{wb -> ConditionalExpression[wa, 0 < wa < 1.]}} is correct. $\endgroup$