From a standard deck of cards ($52$ cards), you and the dealer take turns drawing a card. The dealer draws the card first. The one who draws $8$ diamonds the fastest wins. Calculate the probability of winning.
On the dealer's first move, he has the probability of drawing a card of the desired suit $\frac{13}{52}$ versus $\frac{39}{52}$ the probability of drawing a card of the other suit.
In our first step, two options are possible, depending on what happened at the dealer:
$1)$ if the dealer pulled out a diamond card in the first step: diamond card with probability $\frac{12}{51}$ versus non diamond with probability $\frac{39}{51}$
$2)$ if the dealer did not pull out a diamond card in the first step: diamond card with probability $\frac{13}{51}$ versus non diamond with probability $\frac{38}{51}$
The dealer's next step must take into account the previous state of the system, and so on. Thus, the outcome tree is doubled at each step.
It seems that there is an asymmetry in the game (due to the fact that the dealer goes first) – a favorable outcome of $1$ round of the game for us is only the outcome in which the dealer does not draw a diamond card, but we draw it out. The remaining outcomes are either in favor of the dealer or are neutral. However, I don't see how to scale this idea to the whole game and how to do without calculating the winning paths.
Best Answer
To summarize the discussion in the comments:
Since it is impossible for both players to draw $8$ diamonds, there is no race involved here. You will win if and only if there are $8$ or more diamonds in even positions at the start of the game. Thus the probability that you will win is given by $$\sum_{k=8}^{13} \frac {\binom {13}k \times \binom {39}{26-k}}{\binom {52}{26}}\approx .2615$$
Worth remarking that it would be considerably more work to solve the same problem with $6$ diamonds instead of $8$. In that case, it is possible that both players have enough diamonds in favorable locations to win the game, so the exact locations of the diamonds in the deck becomes relevant.