Simultaneous convoluted linear equations

[Nonlinear_Zero-Sum]

While doing an odds-to-probability derivation, I came upon an interesting relationship between x and y in a pair of simultaneous equations.

Care to solve for x and y..?



Hint: This avenue ended up being a dead-end, derivation-wise.

Thanks,

NLZS

 

[Con-Tester]

Both equations reduce to y = x, so there's no unique solution.

 

[Nonlinear_Zero-Sum]

Bingo. Which is why my 'derivation' died. Thanks for confirming.

For kicks, I did a bunch of substitutions in Microsoft Word's 'Equation' function and ended up with some cool x = y formulas:

 

[Dason]

Sounds like you're a fan of continued fractions https://en.wikipedia.org/wiki/Continued_fraction

 

[Nonlinear_Zero-Sum]

Gee, I didn't know I was until I tripped on one.


But are these x=y relationships really a 'continued fraction', since the only variable (x or y as a-sub-n) is the one on the bottom of the inverted pyramid..? The other a-sub's (0, 1, 2..) are just '1'.

 

[Nonlinear_Zero-Sum]

LINEAR vs NONLINEAR ODDS-TO-PROBS CONVERSION

As initially noted in this thread, in trying to establish the relationship between odds and probabilities through the linear model for a two-outcome event, where Prob.1 + Prob.2 = 100%, I’d reached the above dead-end-but-fun equations, trying to solve for the Odds Ratio (Odds.1/Odds.2).

Modeling the true relationship between odds and probabilities results in a simple nonlinear equation (fractional odds, no house take):

See: http://www.talkstats.com/threads/nonlinear-odds-to-probs-conversion.73716/