In a poker game, the probability of getting a "pair" in five cards is ______. [A pair consists of $2$ cards of the same kind (eg. $2$ kings) and $3$ cards that are different from the kind of the pair (eg. different from kings) and that are all different from each other.]
Step-by-Step Solution
Key Concept: A pair requires selecting one rank for the matching pair, then three completely distinct ranks for the remaining cards, with proper counting of combinations and permutations.
Total ways to choose 5 cards from 52 is $\binom{52}{5} = 2,598,960$. For a pair: (1) Choose the rank for the pair: 13 ways. (2) Choose 2 cards from 4 of that rank: $\binom{4}{2} = 6$ ways. (3) Choose 3 different ranks from remaining 12 ranks: $\binom{12}{3} = 220$ ways. (4) Choose 1 card from each of these 3 ranks: $4^3 = 64$ ways. Total favorable outcomes: $13 \times 6 \times 220 \times 64 = 1,098,240$. Probability = $\frac{1,098,240}{2,598,960} \approx 0.4226 \approx 0.422$.
Correct Answer: 0.422