Question

A pair of aces. What is the probability of drawing two aces in two random draws without replacement from a full deck of cards?

Short answer

The probability is \(\frac{1}{221}\).

Step-by-step solution

- Understand the Problem

Determine the total number of aces and the total number of cards in the deck. A deck has 52 cards and 4 aces.

- Calculate the Probability of Drawing the First Ace

The probability of drawing an ace on the first draw is \(\frac{4}{52}\).

- Calculate the Probability of Drawing the Second Ace

After drawing the first ace, there are now 51 cards left and 3 aces remaining. The probability of drawing a second ace is then \(\frac{3}{51}\).

- Multiply the Probabilities

Multiply the probabilities of both events to get the overall probability of drawing two aces in succession. This is \(\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}\).

Key concepts

card games probability

In probability theory, card games are a fantastic way to explore and understand different probability concepts. Let's delve into how probability works with a traditional deck of cards. A standard deck contains 52 cards, divided into 4 suits (hearts, diamonds, clubs, and spades), each containing 13 cards. Probabilities in card games are usually calculated by considering the ratio of favorable outcomes to the total number of possible outcomes.

In the case of drawing two aces without replacement, we start by recognizing that there are initially 4 aces in the 52-card deck. After the first ace is drawn, there are only 51 cards left, with 3 remaining aces. By calculating the sequential probabilities and multiplying them, you can find the overall chance of a specific sequence occurring during the game.

combinatorics

Combinatorics is a branch of mathematics dealing with counting, combination, and permutation of sets. Understanding basic combinatorics can significantly simplify finding probabilities, especially in scenarios like drawing cards from a deck.

When calculating the probability of drawing two aces in two tries, combinatorics help us determine how many ways we can select the cards. For example, before any card is drawn, there are 52 cards with 4 aces. Once a card is selected, we remove it from our calculations, considering fewer cards and aces remain. This understanding allows us to construct accurate mathematical models for diverse card-drawing problems.

conditional probability

Conditional probability is the probability of an event occurring given that another event has already occurred. In card games, this concept is crucial, especially for events that take place in succession without replacement.

In the problem of drawing two aces consecutively, the probability of the second draw depends on the first draw. Initially, the probability of drawing an ace is \(\frac{4}{52}\). However, after drawing one ace, the conditions change: there are now 51 remaining cards and only 3 aces left. Hence, the probability of the second draw being an ace is \(\frac{3}{51}\).

By multiplying these conditional probabilities, we can calculate the overall probability of drawing two aces in succession: \(\frac{4}{52} \times \frac{3}{51} = \frac{1}{221}\).

deck of cards

A standard deck of cards is a foundational element in probability exercises and card games. It consists of 52 cards divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, numbered from 2 to 10, and includes a jack (J), queen (Q), king (K), and ace (A). Understanding the structure of a deck is essential, as it lays the groundwork for calculating probabilities.

Different problems, like drawing specific cards (such as aces) or combinations of cards, rely heavily on knowing the composition of the deck. For instance, knowing there are exactly four aces in a deck helps us accurately calculate probabilities involving aces. This understanding is crucial for more complex probability exercises and real-world applications.