Let $k \geq 2$ be an integer. At time $n=0,$ we shuffle a deck of $2k$ cards of which $k$ are clubs and $k$ are diamonds. We draw a card each turn without replacing. Denote by $C_{n}$ the number of clubs drawn by the turn $n$ and by $\left(\mathcal{F}_{n}\right)$ the natural filtration of $\left(C_{n}\right)$ Let $M_{n}=\frac{k-C_{n}}{2 k-n}, \quad 0 \leq n \leq 2 k-1$ be the proportion of clubss left in the deck after turn $n$. I could show that $\left(M_{n}\right)$ is a martingale.
now let $S$ be the first time at which the card draw is a club. it is a stopping time. show that the card drawn at time $(S+1)$ is equally likely to be a club or a diamond.
I know I have to use Doob's optional stopping with $M_n$ but I don't know how exactly I should proceed, any help will be greatly appreciated, thanks !
If you know $S=s$ then the probability the $s+1^{\text{th}}$ card is also a club is the same as the probability the last, i.e. $2k^{\text{th}}$, card is a club. Both are $\frac{k-1}{2k-s}$ and the denominator must be positive since $S < 2k$ when $k \ge 2$
If you do not yet know $S$, the probability the next card after the first club is also a club must then be the marginal probability the last card is a club. That is $\frac12$ since you started with an equal number of each type.