Probability of the second heart card If you pull a card and it is a heart, you turn it back and you pull another card. If it is not heart, you pull another card without returning the first card back. What is the probability that the second card is a heart.

Case 1 -
First card is heart = $\frac {13}{52} = \frac {1}{4}$
Second card is heart = $\frac {13}{52} = \frac {1}{4}$
Probability = $\frac {1}{4} × \frac {1}{4} = \frac {1}{16}$
Case 2 -
First card not heart = $\frac {39}{52} = \frac {3}{4}$
Second card is heart = $\frac {13}{51}$
Probability = $\frac {3}{4} × \frac {13}{51} = \frac {13}{68}$
Total probability = $\frac {1}{16} + \frac {13}{68} = \frac {21}{272}$
