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Q: Given 3 decks of cards, A to K. Randomly pick 1 card from each deck. What is the probability that the 3 cards are in increasing order?

A

$$ \mathbb{P}(\text{3 different cards} \cap \text{3 increasing values}) = \mathbb{P}(\text{3 different cards}) \times \mathbb{P}(\text{3 increasing values} | \text{3 different cards}) $$

First, make sure the three cards are different, \(\mathbb{P}(\text{3 different cards})\) is \( 1 * \frac{48}{52} * \frac{44}{52} \). For any 3 cards, the permutation is \(3!\). Given that these 3 cards are different, there is only one increasing order. Thus, \(\mathbb{P}(\text{3 increasing values} | \text{3 different cards}) = \frac{1}{3!}\). Therefore, the result is \(\frac{48 \times 44}{52 \times 52 \times 6} = 0.1318\)