Two cards are drawn successively without replacement from a standard 52‑card deck. What is the probability that the second card is a king given that the first card drawn is a heart?

Step-by-step solution

1. Identify the conditional probability to be found: $P(\text{second is king}\mid \text{first is heart})$.
2. Consider the two possible cases for the first heart: it could be the king of hearts or any other heart.
3. If the first card is the king of hearts (probability $1/13$), 3 kings remain among the 51 cards left. Probability of second being king in this case is $3/51$.
4. If the first card is not the king of hearts (probability $12/13$), 4 kings remain among the 51 cards left. Probability in this case is $4/51$.
5. Apply the law of total probability: $P=\frac{1}{13}\cdot \frac{3}{51}+\frac{12}{13}\cdot \frac{4}{51}=\frac{3+48}{663}=\frac{51}{663}=\frac{1}{13}$.
6. Thus the required probability is $\boxed{1/13}$.

Concept tested: Conditional probability with dependent draws

Published 18 September 2026 · How this solution was prepared and checked