Question

A card is drawn from a well-shuffled pack of 52 eards, then the probibility of this card being a red coloured ace is.

Short answer

The probability is \( \frac{1}{26} \).

Step-by-step solution

Understand the Problem

We need to find out the probability of drawing a red ace from a standard 52-card deck. A standard deck includes 52 cards, with 26 red cards and 26 black cards. Red cards consist of hearts and diamonds.

Identify the Total Number of Red Aces

In a deck of 52 cards, there are two red aces: the Ace of Hearts and the Ace of Diamonds.

Determine the Total Number of Possible Outcomes

The deck consists of 52 cards, so the total number of possible outcomes when drawing a single card is 52.

Calculate the Probability

The probability of drawing one of the red aces is calculated by dividing the number of favorable outcomes by the total number of possible outcomes: \[ P(\text{red ace}) = \frac{2}{52} = \frac{1}{26} \]

Key concepts

Card Probability

The concept of card probability involves determining the likelihood of drawing a specific card or set of cards from a deck. A standard deck consists of 52 cards split into four suits: hearts, diamonds, clubs, and spades.
Out of these, hearts and diamonds are red, while clubs and spades are black. When calculating probabilities, it helps to differentiate between these suits.
For instance, in the case of the exercise, we specifically want to determine the probability of drawing a red ace.

• There are 13 cards per suit in a deck.
• Red suits include hearts and diamonds.
• Each suit contains one ace, making two red aces in total.

Understanding these basic compositions of a deck is crucial in card probability scenarios.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting and arranging objects. It plays a significant role in probability, especially in problems involving cards.
All possible arrangements and combinations of drawing cards can be quantified and categorized with combinatorial techniques.
In card games, combinatorics can be used to calculate how many specific card combinations are possible, such as pairs, flushes, or in this case, red aces.

• "Combinations" involve selecting items irrespective of order.
• Each draw of a card is independent, meaning previous draws do not influence future draws.
• Combinatorics help isolate favorable outcomes, like finding instances of red aces.

By using combinatorial approaches, one can efficiently determine how many favorable scenarios or combinations exist in a problem.

Probability Calculation

Probability calculation is determining how likely an event is to occur. It finds extensive applications in card games.
The formula used is basic probability: dividing the number of favorable outcomes by the total number of possible outcomes.
In the context of a deck of cards, it looks like this:

• "Favorable outcomes" are the particular outcomes we're interested in, such as drawing a red ace, which are 2 in total.
• "Possible outcomes" denote all outcomes that can occur, like the 52 cards in the deck.
• The probability is formulated as: \( P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).

For the exercise, since there are 2 red aces out of 52 cards, the probability is computed as \( \frac{2}{52} = \frac{1}{26} \), showing how likely you are to draw a red ace when pulling one card from the deck.