So I have two known points, $A < B$, on the $x$-axis of the normal CDF with unknown mean $\mu$ and unit standard deviation.
The conditional probability: $$P = \Pr\left( x < A \mid x < A \lor x > B \right)$$
is known and given.
I am hoping to determine how to come up with an analytic estimate of the mean $\mu$ in the following equation
$$(1 - P) \times \Phi(\mu-B) = P \times \Phi(A-\mu)$$
where $\Phi$ is the standard normal CDF, and points $A$ and $B$ and the conditional probability $P$ are all constants.
Any solutions, suggestions, or shoves in the right direction would be greatly appreciated.
EDIT:
The best I have come up with is to use Newton's method to estimate the value of $\mu$ where:
$$(1 - P) \times \Phi(\mu-B) - P \times \Phi(A-\mu) = 0$$
For my business domain I have found the following to be a sufficiently reliable initial estimate for $\mu$:
$$x_0 = \frac{A + B}{2} + \Phi^{-1}(P)$$
And then using Newton's method the estimate at the $n^{th}$ iteration is:
$$x_n = x_{n-1} + \frac{P \times \Phi(A-x_{n-1}) - (1 - P) \times \Phi(x_{n-1}-B )}{P \times \phi(A-x_{n-1}) + (1 - P) \times \phi(x_{n-1}-B)}$$
where $\phi$ is the standard normal PDF.
I suppose this is more of a workaround then an actual answer, but for my purposes it does the trick.