Understanding Probability in Playing Cards
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur. When working with a standard deck of 52 cards, we use the basic formula:
$$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
The Problem: King or Diamond
We want to find the probability of drawing a card that is either a King or a Diamond. This is a classic example of the Addition Rule for Probability:
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
Step-by-Step Solution
- Identify the Sample Space ($S$): A standard deck has $n(S) = 52$.
- Identify Event A (Drawing a King): There are 4 Kings in a deck (Hearts, Diamonds, Clubs, Spades). So, $n(A) = 4$.
- Identify Event B (Drawing a Diamond): There are 13 Diamonds in a deck (Ace through King of Diamonds). So, $n(B) = 13$.
- Identify the Overlap ($A \cap B$): Is there a card that is both a King and a Diamond? Yes, the King of Diamonds. So, $n(A \cap B) = 1$.
- Apply the Formula: $$P(A \cup B) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52}$$ $$P(A \cup B) = \frac{16}{52}$$
- Simplify: Divide numerator and denominator by 4: $\frac{16 \div 4}{52 \div 4} = \frac{4}{13}$.
Intuition
When you count the 4 Kings and the 13 Diamonds, you count the 'King of Diamonds' twice. By subtracting the intersection ($A \cap B$), we ensure that the overlap is only counted once, leading to the correct probability of $4/13$.