[(n-1)/n]^n also equals 1/e for large n, so in this problem the cards are independent, and we can just do (51/52)^52 = 36.4% without the inclusion-exclusion principle.


If you do something n times where the probability of success each time is 1/n, and the tries are independent, then the probability of at least one success is approximately 1-1/e = 63%, the approximation becoming better as n becomes sufficiently large.