I have deck of 52 shuffled cards from which I draw cards one by one and an indicator $I_k$ where $ I_k \in \{0, 1\}$ that shows the k-th card is a King or not. My filtration is $\mathcal{F}_n = \sigma(I_1, I_2,...,I_n)$. I also have a probability estimate of the last card being a King $G_n = \mathbb{E}(I_{52}| \mathcal{F}_n)\;\;\forall n \in$ {0,1,...,51}. I wish to know how to express $G_n$ in terms of $I_1,I_2,...I_n$. I have tried solving it myself as follows:
Number of kings drawn after n draws would be $\sum^n_{k=1} I_k$. The remaining number of kings would be $4 - \sum^n_{k=1} I_k$. Since the remaining 52 - n cards have equal chance to be a king, then $\mathbb{E}(I_{52}| \mathcal{F}_n) = \frac{4 - \sum^n_{k=1} I_k}{52 - n}$.
Secondly I would also like to know how to show if $G_n$ is a martingale.
Thank you in advance
The formal proof that $G_n$ is a martingale is just applying the tower property: Since $\mathcal F_{n-1} \subset \mathcal F_{n}$, we have \begin{align*} \mathbb{E}[G_{n}|\mathcal F_{n-1}] &= \mathbb{E}\big[\mathbb{E}[I_{52}|\mathcal F_{n}]\big|\mathcal F_{n-1}\big] = \mathbb{E}[I_{52}|\mathcal F_{n-1}] = G_{n-1}. \end{align*}
Your computation of $\mathbb{E}[I_{52}|\mathcal F_{n}]$ looks correct.