Suppose that an ordinary deck of 52 cards (which contains 4...

Suppose that an ordinary deck of 52 cards (which contains 4 aces) is randomly divided into 4 hands of 13 cards each. We are interested in determining $p$, the probability that each hand has an ace. Let $E_{i}$ be the event that the $i$ th hand has exactly one ace. Determine $p=P\left(E_{1} E_{2} E_{3} E_{4}\right)$ by using the multiplication rule.

Key Concepts

Sampling Without Replacement

Sampling without replacement refers to the process of drawing items from a set where each item is not returned after being drawn, which alters the probabilities for subsequent draws. This concept is crucial in card dealing problems since the composition of the deck changes as cards are dealt to each hand, affecting the calculations of conditional probabilities.

Combinatorial Analysis

Combinatorial analysis involves counting and arranging items to determine the total number of possible outcomes versus the number of favorable outcomes. In the context of dealing cards, it provides the tools to count the ways to distribute specific cards (such as aces) among the hands, which is essential for calculating probabilities in this context.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. In problems involving card dealing without replacement, the probability of a card being an ace in a subsequent hand depends on the previous hands’ outcomes, making conditional probability a key concept for accurately determining overall probabilities.

Multiplication Rule in Probability

The multiplication rule allows us to calculate the probability of multiple events all occurring by multiplying together the probability of the first event with the conditional probabilities of each subsequent event given that the previous events have occurred. This is particularly useful in sequential or dependent event scenarios, such as dealing cards where the outcome of one deal affects the probabilities in the next.

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