I am confident there is a very well known solution to this, and yet I am having a lot of trouble finding the relevant reference. If you could take a quick look, it would be very much appreciated!
TL;DR Version: Suppose $N$ iid Bernoulli trials are performed. Further suppose that the number of successes by the $k^{\text{th}}$ trial have been observed. What is the probability of $m\in\{k,...,\ell\}$ successes by the trial $\ell\in\{k+1,...,N\}$, as calculated by Bayes' rule?
Formal Version: First let $\varphi\in(0,1)$ and define $$f(a|b) = \begin{cases} \varphi, &\text{if } a=b+1\\ 1-\varphi, &\text{if } a=b\\ 0, &\text{otherwise} \end{cases}, \quad (b = 0,1,2,...) $$
Now, consider a random vector $\boldsymbol{S}=(S_1,...,S_N)\in\{0,1\}^N$ with a probability mass function $\mu(\cdot)$ given by:
$$ \mu(\boldsymbol{s}) = \left[\prod_{n=2}^{N}f(s_n|s_{n-1})\right]\varphi^{s_1} (1-\varphi)^{1-s_1}, \qquad (\boldsymbol{s}=(s_1,...,s_N)\in\{0,1\}^N).$$
Let $k\in\{1,...,N-1\}$ and $\ell\in\{k+1,...,N\}$. Then, how would I calculate $\text{Prob}\{S_\ell = s_{\ell}|S_k = s_k\}$, where $s_k\in\{1,...,k\}$ and $s_\ell\in\{k,...,\ell\}$?
Thank you very much!