We play a game with a (shuffled) standard 52 poker cards deck where each card value is in $\{A,2,3,4,5,6,7,8,9,10,J,Q,K\}$. For simplicity, relabel the 13 values as $\{1,2,3,4,5,6,7,8,9,10,11,12,0\}$. The game requires each turn to not flip the card with the turn's number as value. Formally:
Game rules:
i. At each turn $n\ge 1$ we flip a card.
ii. If at turn $n$ we flip $n$ mod $13$ we lose, otherwise we move on to turn $n+1$.
iii. We win if we reach turn $n=52$ (i.e. we flip all cards in the deck) and the last card is not a $0$ (i.e. a $K$).
Question: What is the probability of winning?
(Comment: actually I am struggling with the same game with, say, a deck of 10 cards with values $\{1,2,3,4,5,6,7,8,9,0\}$. I tried to calculate the number of deck's permutations such that the $i$-th position does not contain value $i$, but I can't find a simple way to count this.)