A single card is drawn at random from a shuffled deck. What is the probability that it is red? That it is the ace of hearts? That it is either a three or a five? That it is either an ace or red or both?

Probability is the measure of the chance that a particular event will occur. To calculate the probability of an event, you use the formula:

\( P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

This formula gives you the likelihood of drawing a particular card from a deck, selecting a specific outcome, or any other similar event.
For example, to find the probability of drawing a red card from a standard 52-card deck, identify the number of red cards (26) and divide it by the total number of cards (52). This yields:
\[ P(\text{Red card}) = \frac{26}{52} = \frac{1}{2} \]
Knowing how to calculate probability enables you to understand and predict outcomes in various scenarios.

A standard deck of cards consists of 52 cards divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards: Ace through King.

• There are 2 red suits: hearts and diamonds
• There are 2 black suits: clubs and spades
• Each suit has 1 Ace, 1 King, 1 Queen, 1 Jack, and number cards from 2 to 10

This structure is essential when calculating probabilities involving cards. For example, knowing there are 13 hearts allows you to calculate the probability of drawing the ace of hearts. Since there is only one ace of hearts in the deck, the probability is:
\[ P(\text{Ace of hearts}) = \frac{1}{52} \]
Being familiar with the deck’s structure makes these calculations straightforward and intuitive.

The inclusion-exclusion principle is a technique used to calculate the probability of the union of multiple events. When events overlap (like being an ace and being red), simple addition of their probabilities would double-count certain outcomes. The formula is:
\[ P(A \bigcup B) = P(A) + P(B) - P(A \bigcap B) \]
This approach corrects for the over-counted intersections.
For example, to find the probability of drawing a card that is either an ace or red or both, calculate each separately, sum them, and subtract the overlap where they intersect (red aces):
\[ P(\text{ace} \bigcup \text{red}) = \frac{4}{52} + \frac{26}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13} \]
This corrects the double count and gives you the accurate probability.

Combinatorics involves counting, arranging, and combining objects effectively. It is the foundation for calculating probabilities. Common methods include permutations and combinations. In card games, understanding the number of ways to draw certain cards (favorable outcomes) helps determine probabilities.

For instance, consider the calculation for the probability of drawing either a three or a five. There are 4 threes and 4 fives, giving 8 favorable outcomes. The total number of possible outcomes being 52 cards:
\[ \frac{8}{52} = \frac{2}{13} \]
Basic combinatorics allows you to break down complex problems into simpler, manageable counts, making probability calculations easier.