I have 3 formulae used to calculate some probabilities of some events, 0, 1, 2 occurring.
P($X=0$) = $(1-$z$)^R$ $\cdot$ $z^A$
P(X = 1) = $0.5^{R+A}$
P(X = 2) = $z^R$ $\cdot$ $(1-$z$)^A$
Where $z$ = 0.01, and $P(X=0) + P(X=1) + P(X=2) = 1$
$R$ and $A$ were known originally and used to calculate the values of $P(X=0), P(X=1), P(X=2)$.
I have the set of values corresponding to $P(X=0), P(X=1), P(X=2)$, for example, 0.660 0.333 0.007. However, the $R$ and $A$ used to calculate each trio have since been 'lost'. For each trio, I would like to know the values of $R$ and $A$ given $P(X=0), P(X=1), P(X=2)$.
I suppose this involves solving 2 simultaneous equations:
$0.99^R$ $\cdot$ $0.01^A$ = $P(X=0)$
$0.01^R$ $\cdot$ $0.99^A$ = $P(X = 2)$
I am unsure if this can be achieved analytically. Can anyone help?