Two cards are drawn from a well-shuffled, standard deck of...

Two cards are drawn from a well-shuffled, standard deck of playing cards [four suits (diamonds, hearts, clubs, and spades) of 13 cards each (ace through king)] If the first card is not replaced between selections, then what is the probability that: $(a)$ both cards will be hearts, (b) both will be queens, (c) one will be a king and the other a queen?

Two cards are drawn from a well-shuffled, standard deck of playing cards [four suits (diamonds, hearts, clubs, and spades) of 13 cards each (ace through king)] If the first card is not replaced between selections, then what is the probability that: $(a)$ both cards will be hearts,
(b) both will be queens, (c) one will be a king and the other a queen?

Key Concepts

Standard Deck Composition

A standard deck of cards is a fundamental concept in probability that consists of a fixed number of cards, typically 52, divided equally among four suits. Each suit contains the same number of cards, providing a structured and balanced sample space that is often used to illustrate probability principles and combinatorial analysis.

Dependent Events

When selections are made without replacement, the outcome of the first selection impacts the probabilities of subsequent selections. This dependency means that the probability of each event must account for the change in the sample space, which is a key consideration in calculating probabilities for sequential events.

Conditional Probability

Conditional probability is a concept that quantifies the likelihood of an event occurring given that another event has already occurred. In scenarios where cards are drawn without replacement, the outcome of the first draw alters the conditions for the next draw, requiring the use of conditional probability to accurately determine the overall likelihood of a particular sequence of events.

Multiplication Rule for Dependent Events

The multiplication rule for dependent events is used to calculate the probability of two sequential events occurring jointly. It involves multiplying the probability of the first event by the conditional probability of the second event given that the first event has occurred, which is crucial when the events do not have independent probabilities.

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