Before drawing the first card, you have 0 values which occur 0 times in the stack, 0 values which occur 1 time, ..., 13 values which occur 4 times. This can be written as (0,0,0,0,13).
You guess a random value for the first card. With 1/13 probability, the card has this value, and you lose. Otherwise, some other value has only 3 cards left, which gives you (0,0,0,1,12).
Now one value is less frequent than the others, therefore you will guess this value.
How do these numbers develop? Can you calculate the probability, based on this development?
This might need some computer assistance. And store your calculated intermediate values to re-use them.
I get ~27% as result.