You are dealt a hand consisting of 5 cards from a standard...

You are dealt a hand consisting of 5 cards from a standard deck of 52 cards. Determine the probability of obtaining the following hands:
a. flush (five cards of the same suit)
b. a king, queen, jack, ten, and ace of the same suit (a "royal flush")

Key Concepts

Combinatorics and Counting Principles

This concept involves using methods such as permutation and combination to count the number of ways items can be selected or arranged. In problems involving card hands, combinations are used to determine both the total number of possible hands and the number of favorable hands that meet specific conditions.

Probability Theory

Probability theory quantifies the likelihood of an event occurring by comparing the number of favorable outcomes to the total number of outcomes. When applied to card games, it involves calculating ratios that express the chance of drawing a hand with specific attributes from a known set of possibilities.

Standard Deck of Cards

A standard deck of cards consists of 52 cards divided evenly among four suits (hearts, diamonds, clubs, and spades), with each suit containing 13 ranks. This structure is fundamental in determining the total sample space and the specific counts for various card combinations in probability problems.

Flush

A flush is a hand where all cards are of the same suit, regardless of their ranks. Recognizing a flush in probability problems requires counting the number of ways to select cards such that they all belong to one suit and comparing that to the total number of possible hands.

Royal Flush

A royal flush is a highly specific type of flush that consists of the 10, jack, queen, king, and ace of a single suit. It is a rare event in card games, and its probability is determined by identifying the single possible combination per suit and comparing it to the total number of hands.

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