It may be an obvious question, but how can we correctly evaluate, in general, the time to first success in a process in which the probability increases at each trial?
The particular example I have in mind is the event $L_k$ "to get, in $k$ independent trials (i.e. with replacement), at least one element of kind A and at least one element of kind B", taken from an urn containing $\alpha$ elements of kind A, $\beta$ elements of kind B and $\gamma$ elements of kind G, and $c=\alpha+\beta+\gamma$. Clearly, $$P(L_k)=1-\left(\frac{\alpha+\gamma}{c}\right)^k-\left(\frac{\beta+\gamma}{c}\right)^k+\left(\frac{\gamma}{c}\right)^k,$$
which (given $\alpha,\beta,\gamma>0$) is a strictly monotonically increasing function of $k$. Then, the question is: If we perform $k=n$ independent trials, what is the expected number of trials to first success for the event $L_n$?
This a useful trick for computing expected values of integer-valued random variables: $$ E[X] = \sum_{i=0}^\infty P(X>i) $$ In your case, let $X$ be number of trials it takes to get at least one of A and at least one of B. Then$$P(X>i)=1-P(L_i)=\left(\frac{\alpha+\gamma}{c}\right)^i+\left(\frac{\beta+\gamma}{c}\right)^i-\left(\frac{\gamma}{c}\right)^i,$$ so using the formula for the sum of an infinite geometric series, $$ E[X]=\sum_{i=0}^\infty\left(\frac{\alpha+\gamma}{c}\right)^i+\left(\frac{\beta+\gamma}{c}\right)^i-\left(\frac{\gamma}{c}\right)^i=\frac{c}{\beta}+\frac{c}{\alpha}-\frac{c}{\alpha+\beta} $$