Two cards from a pack of $52$ cards are lost. One card is drawn from the remaining cards. If drawn card is heart, find the probability that the lost cards were both hearts.
Do we have to define events for all cases like:
Missing cards are heart and diamond or Missing card are two hearts etc and then apply Bayes Theorem or is there a better approach to solve this?
Best Answer
Having one heart in hand, there are $\binom{51}{2}$ ways to choose to missing cards, and $\binom{12}{2}$ ways to choose two missing hearts. $$\frac{\binom{12}{2}}{\binom{51}{2}}=\frac{66}{1225}\approx 0.051765$$ This is equivalent to drawing three cards from a full deck: Given that the first card is a heart, find the probability that the second and third cards are also hearts. $$\frac{12}{51}\cdot\frac{11}{50}=\frac{132}{2550}\approx 0.051765$$