A deck of playing cards contains 52 cards, four of which are...

A deck of playing cards contains 52 cards, four of which are aces. What is the probability that the deal of a five-card hand provides:
a. A pair of aces?
b. Exactly one ace?
c. No aces?
d. At least one ace?

Key Concepts

Complement Rule in Probability

The complement rule is a fundamental strategy in probability that involves finding the probability of an event by subtracting the probability of its complement from one. It simplifies calculations, especially when it is easier to count the outcomes for the event not happening, such as determining the probability of drawing at least one ace.

Hypergeometric Distribution

This distribution models the probability of drawing a specific number of successes from a finite population without replacement. It is particularly relevant to card problems where the draws are dependent, such as calculating the probability of drawing a certain number of aces from a deck.

Combinatorial Counting

This concept involves calculating the number of ways to choose a subset of items from a larger set without considering the order. In probability problems like card hands, it is used to determine both the total number of possible outcomes and the number of favorable outcomes by applying counting principles.

Combination Formula (Binomial Coefficients)

The combination formula, represented as nCk, is essential for finding the number of ways to select k items from a set of n. It plays a crucial role in probability problems by providing the mechanism to compute the number of ways specific selections (like choosing a certain number of aces) can occur.

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