Given:
p = $a^l$ * ($\frac{c}{a^l + b^l}$ + $\frac{b}{a^l + c^l}$)
Where a, b, c, p are known and are probabilities.
Solve for l. (1 equation and 1 unknown)
Does a closed form solution to this exist? I can't see how to solve using algebra.
If there is no closed form solution, what is the best way to solve this? I have looked into using Newton's method (but the derivative of this is very ugly) and the secant method (I am concerned about using bad starting values) - these aren't very "elegant".
Edit
Working on this problem, I have found out the following constraints for solutions:
- $a$ >= $b$ >= $c$
- $a$ + $b$ + $c$ = $1$
- $p$ <= $(1 - a)$
I believe that the upper and lower limits of $l$ (that I could use as inputs into the secant method) for the given $a$, $b$, $c$, $p$ are defined by:
- lower: $p$ >= 0
- upper: $p$ <= $(1 - a)$