Two cards are drawn at random from a shuffled deck. (a) What...

Two cards are drawn at random from a shuffled deck.
(a) What is the probability that at least one is a heart?
(b) If you know that at least one is a heart, what is the probability that both are hearts?

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Key Concepts

Combinatorial Analysis

Combinatorial analysis involves techniques for counting and enumerating possible outcomes in a discrete sample space. This includes using combinations and permutations to determine the total number of ways in which events can occur, which is essential for calculating probabilities in scenarios with multiple draws or selections.

Complementary Probability

Complementary probability is a method used to calculate the likelihood of an event by subtracting the probability of its complement from one. This approach is often useful when it is simpler to determine the probability that an event does not occur rather than calculating the event's probability directly.

Conditional Probability

Conditional probability concerns the likelihood of an event occurring given that another event has already occurred. This concept is crucial for updating probabilities based on new information and is typically represented by the formula P(A|B) = P(A and B) / P(B).

Sampling Without Replacement

Sampling without replacement refers to the process of drawing items from a finite set where each item is not returned to the set after being drawn. This affects the probability of subsequent events, as the sample space is reduced after each draw, making it an important consideration in many probability problems.

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