What is the probability that it will take n draws to draw 2 cards that are the same suit in a deck of 52 cards??

Question

Two cards are drawn at random from a well-shuffled standard deck of 52 cards. If the two cards are the same suit, we stop. If they are from different suits, the cards are returned to the deck, the deck reshuffled, and the process is repeated. What is the probability that it will take n draws until we stop?

Answer 1

Each time you draw two cards from the deck, the probability that the cards are not of the same suit is $39/51$ and the probability that the cards are from the same suit is $12/51$.

Since the processes of drawing two cards are mutually independent, the probability that we stop at the n'th process is $(39/51)^{n-1}(12/51)$.

Answer 2

The probability of having the same suit the first time is:

$$\binom{4}{1}\frac{\binom{13}{2}}{\binom{52}{2}}$$

Now say we didnt get the same suit first time AND=* we get the same suit the second time:

$$\binom{4}{1}\frac{\binom{13}{1}\binom{39}{1}}{\binom{52}{2}}*\binom{4}{1}\frac{\binom{13}{2}}{\binom{52}{2}}$$

If we do this for n tries :

$$\binom{4}{1}\sum_{i=1}^n(\frac{\binom{13}{1}\binom{39}{1}}{\binom{52}{2}})^{i-1}*(\frac{\binom{13}{2}}{\binom{52}{2}})$$