In the case of the binomial distribution, it is well known that given $N$ trials, the expected number of trials to get a success is $Np$, where $p$ is the probability of success, and the variance is $Npq$, where $q = 1-p$.
My question is : what if the probability is time-dependent how do these results change ? Is there some formula for the average and standard deviation? Are there any references ?
You can work out the maths by yourself : let $X_k$ be a Bernoulli trial with success probability $p_k$. Now let $S_n=X_1+...+X_n$. What you're looking for is $\mathbb{E}[S_n]$. It is given the same as in the usual case : $$\mathbb{E}[S_n]=\mathbb{E}[X_1]+...+\mathbb{E}[X_n]=p_1+...+p_n.$$
The Binomial Distribution is a special case where $p_1=...=p_n=p$, yielding $\mathbb{E}[X_n]=np$. The same goes for the variance, assuming all $X_k$ are independant : $$\mathbb{V}(S_n)=\mathbb{V}(X_1)+...+\mathbb{V}(X_n)=p_1(1-p_1)+...+p_n(1-p_n).$$
Once again, the Binomial Distribution is a special case, and the corresponding formula holds.
However, if you're interested in non-independant behaviour (e.g. if the probability $p_{k+1}$ depends on the result $X_k$), it gets a little more complicated. I suggest you take a look at Conditional probabilities first.
Edit : as suggested by
fGDu94, I'll link the Poisson Binomial Distribution as well for further documentation.