A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a king or a club?

Answer

To find the probability of choosing a king or a club from a standard deck of 52 playing cards, we will follow these steps:

### Step 1: Identify the total number of outcomes
A standard deck of playing cards has 52 cards. Therefore, the total number of possible outcomes when choosing one card is:
\[ \text{Total outcomes} = 52 \]

**Hint:** Remember that the total number of cards in a standard deck is always 52.

### Step 2: Count the number of favorable outcomes for kings
In a deck, there are 4 kings (one from each suit: hearts, diamonds, clubs, and spades). Thus, the number of favorable outcomes for choosing a king is:
\[ \text{Favorable outcomes for kings} = 4 \]

**Hint:** Think about how many suits there are and how many kings are in each suit.

### Step 3: Count the number of favorable outcomes for clubs
There are 13 clubs in a deck (numbered 2 through 10, plus the jack, queen, king, and ace). Therefore, the number of favorable outcomes for choosing a club is:
\[ \text{Favorable outcomes for clubs} = 13 \]

**Hint:** Recall that each suit has 13 cards.

### Step 4: Count the overlap (intersection) of kings and clubs
Since one of the kings is also a club (the king of clubs), we need to account for this overlap. Thus, the number of favorable outcomes that are both a king and a club is:
\[ \text{Favorable outcomes for king of clubs} = 1 \]

**Hint:** Consider how many cards belong to both categories (kings and clubs).

### Step 5: Apply the formula for the probability of the union of two events
To find the probability of choosing a king or a club, we use the formula for the union of two events:
\[ P(K \cup C) = P(K) + P(C) - P(K \cap C) \]
Where:
- \( P(K) \) is the probability of choosing a king,
- \( P(C) \) is the probability of choosing a club,
- \( P(K \cap C) \) is the probability of choosing the king of clubs.

Calculating each part:
- \( P(K) = \frac{4}{52} \)
- \( P(C) = \frac{13}{52} \)
- \( P(K \cap C) = \frac{1}{52} \)

Substituting these values into the formula:
\[ P(K \cup C) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} \]
\[ P(K \cup C) = \frac{4 + 13 - 1}{52} = \frac{16}{52} \]

### Step 6: Simplify the probability
Now, we simplify \( \frac{16}{52} \):
\[ P(K \cup C) = \frac{16 \div 4}{52 \div 4} = \frac{4}{13} \]

### Final Answer
The probability of choosing a king or a club is:
\[ \frac{4}{13} \]

---