I have a very simple problem.
Lets assume that I have a well shuffled deck of 52 cards. I start drawing the top card always and when the card matches its rank I lose. J=11 Q=12 K=13.
If there were only 13 cards I could easily use the $\ \frac{!n}{n!}$ for derangements in order to solve this. The problem is that there are 52 cards so when I pass 13 I start from 1 again so I don't know what is the probability to win. Example of the game
1st card: 4 - Continue
2nd Card: A - continue
3rd Card: K - Continue
4th Card: K - Continue
5th Card: 6 - Continue
6th Card: 9 - Continue
7th Card: 10 - Continue
8th Card: A - Continue
9th Card: J - Continue
10th Card: 3 - Continue
11th Card: 2 - Continue
12th Card: 8 - Continue
13th Card: A - Continue
1st card: 5 - Continue
2nd Card 2 LOSE
So actually I have to count from 1 to 13 4 times and if I draw all 52 cards then I win. What's the probability?