Two of a Kind Follow the outline presented in Problem 67 to determine the probability of being dealt exactly one pair.

Verified step by step guidance

1. Understand the problem: We want to find the probability of being dealt exactly one pair in a 5-card hand from a standard 52-card deck. "Exactly one pair" means two cards of the same rank and the other three cards all of different ranks, none matching the pair's rank.
2. Calculate the total number of 5-card hands possible from a 52-card deck using combinations: \(\binom{52}{5}\).
3. Determine the number of ways to choose the rank for the pair: there are 13 possible ranks, so choose 1 rank out of 13, which is \(\binom{13}{1}\).
4. For the chosen rank, select 2 cards out of the 4 available suits to form the pair: \(\binom{4}{2}\).
5. For the remaining 3 cards, select 3 different ranks out of the remaining 12 ranks (since one rank is used for the pair): \(\binom{12}{3}\). For each of these ranks, select 1 card out of 4 suits, so multiply by \$4^3$.
6. Multiply all these counts together to get the number of favorable hands, then divide by the total number of 5-card hands to get the probability.

Key Concepts

Probability of a Specific Event

This concept involves calculating the likelihood of a particular outcome occurring within a defined sample space. In card problems, it requires identifying all favorable outcomes and dividing by the total number of possible outcomes, often using combinatorial methods.

Combinatorics and Counting Techniques

Combinatorics deals with counting the number of ways to select or arrange items. For card hands, combinations (nCr) are used to count possible card selections without regard to order, essential for determining the number of hands with exactly one pair.

Definition of Exactly One Pair in Poker Hands

Exactly one pair means having two cards of the same rank and the other three cards all of different ranks, none matching the pair. Understanding this helps in correctly counting favorable hands and avoiding overcounting or undercounting.