Understanding Probability
Probability is the branch of mathematics that quantifies the likelihood of an event occurring. At its core, the probability of an event $E$ is given by the simple ratio:
$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
The Anatomy of a Deck of Cards
A standard deck of playing cards contains 52 cards. These are divided into four suits:
- Spades (♠)
- Hearts (♥)
- Diamonds (♦)
- Clubs (♣)
Each suit consists of 13 ranks: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King.
Solving the Problem
Question: A card is drawn at random from a well-shuffled deck of 52 cards. What is the probability that it is a spade?
Step 1: Identify the total number of outcomes. There are 52 cards in a deck, so the total number of possible outcomes is 52.
Step 2: Identify the number of favorable outcomes. Since there are four suits and each suit has 13 cards, the total number of spades is 13.
Step 3: Apply the probability formula. $$P(\text{Spade}) = \frac{13}{52}$$
Step 4: Simplify the fraction. Dividing both the numerator and the denominator by 13, we get: $$P(\text{Spade}) = \frac{1}{4}$$
Intuition
Since there are four suits and the deck is "well-shuffled," each suit has an equal chance of being selected. Because there are 4 suits, the chance of picking any specific suit is $1/4$ or 25%.