Probability Basics with Playing Cards
Probability is the study of how likely an event is to occur. When working with a standard deck of cards, we are dealing with a set of $n(S) = 52$ equally likely outcomes.
Solving the Problem
Question: What is the probability of drawing a King or a Queen from a well-shuffled deck of 52 cards?
Step 1: Identify the Sample Space
The total number of cards in the deck is $52$. Thus, $n(S) = 52$.
Step 2: Define the Events
Let $A$ be the event of drawing a King. There are $4$ Kings in a deck (Spades, Hearts, Diamonds, Clubs). So, $n(A) = 4$. Let $B$ be the event of drawing a Queen. There are $4$ Queens in a deck. So, $n(B) = 4$.
Step 3: Check for Overlap
Events $A$ and $B$ are mutually exclusive because a single card cannot be both a King and a Queen at the same time. Therefore, $n(A \cap B) = 0$.
Step 4: Apply the Addition Rule
The probability of drawing a King or a Queen is given by the formula: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
$$P(A) = \frac{4}{52}$$ $$P(B) = \frac{4}{52}$$
$$P(A \cup B) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52}$$
Step 5: Simplify
Simplifying $\frac{8}{52}$ by dividing both numerator and denominator by $4$, we get: $$\frac{8 \div 4}{52 \div 4} = \frac{2}{13}$$
Conclusion: The probability of drawing a King or a Queen is $\frac{2}{13}$.