I beg your pardon for this (likely) silly question.
An urn contains $c$ elements of three different types, say $\alpha>0$ elements of type $A$, $\beta>0$ elements of type $B$ and $\gamma>0$ elements of type $G$.
We define the event $L_n^{AB}$ as to get, in $n$ independent trials (i.e. with replacement), at least one element of kind $A$ and at least one element of kind $B$, so that $P(L_n^{AB})=1-\left(\frac{\alpha+\gamma}{c}\right)^n-\left(\frac{\beta+\gamma}{c}\right)^n+\left(\frac{\gamma}{c}\right)^n$.
Clearly, the event $L_n^{AB}$ can occur at each trial, except the first one.
Following the suggestion of the (clever & kind) user Youem, I got the definition
If you have an event $A$, and a number of trials $N$ (which can be $\infty$), then the definition of the "time to first success" of A is $$T_A = \min \{n \le N\;:\; A\;\text{is true at the $n-$th trial}\}$$
But I don't understand how to apply it in the case of the event $L_n^{AB}$.
Many thanks for your help!
My (partial and wobbly) solution is: $T_{L_n^{AB}}=2$ with probability $\frac{2\alpha\beta}{c^2}$ and $T_{L_n^{AB}}>2$ with probability $1-\frac{2\alpha\beta}{c^2}>0$, but it is not really a satisfactory answer.