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Let $K$ be the event that the lost card is a King. Let $E$ be the event that a card drawn from the remaining cards is a King.
We want to find $P(K|E)$, the probability that the lost card is a King given that a card drawn is a King. $P(K) = \frac{4}{52} = \frac{1}{13}$ (Probability that the lost card is a King) $P(K') = 1 - P(K) = 1 - \frac{1}{13} = \frac{12}{13}$ (Probability that the lost card is not a King)
$P(E|K) = \frac{3}{51}$ (Probability of drawing a King given that the lost card was a King) $P(E|K') = \frac{4}{51}$ (Probability of drawing a King given that the lost card was not a King)
Using Bayes' Theorem, we have: $$P(K|E) = \frac{P(E|K)P(K)}{P(E|K)P(K) + P(E|K')P(K')}$$ $$P(K|E) = \frac{\frac{3}{51} \cdot \frac{1}{13}}{\frac{3}{51} \cdot \frac{1}{13} + \frac{4}{51} \cdot \frac{12}{13}}$$ $$P(K|E) = \frac{\frac{3}{51 \cdot 13}}{\frac{3}{51 \cdot 13} + \frac{48}{51 \cdot 13}}$$ $$P(K|E) = \frac{3}{3 + 48} = \frac{3}{51} = \frac{1}{17}$$
Final Answer: 1/17