What is the probability that a card drawn at random from a pack of 52 cards is either a king or a spade?

Options

A. `17/52`
B. `4/13`
C. `3/13`
D. `13/52`

Answer

To find the probability that a card drawn at random from a pack of 52 cards is either a king or a spade, we can follow these steps:

### Step 1: Identify the total number of cards
A standard deck of cards has a total of 52 cards.

**Hint:** Remember that a standard deck consists of 4 suits: hearts, diamonds, clubs, and spades, each containing 13 cards.

### Step 2: Determine the number of kings
There are 4 kings in a deck of cards (one from each suit).

**Hint:** Each suit has one king: King of Hearts, King of Diamonds, King of Clubs, and King of Spades.

### Step 3: Determine the number of spades
There are 13 spades in a deck of cards.

**Hint:** Spades are one of the four suits, and each suit has 13 cards.

### Step 4: Identify the overlap (intersection)
Since one of the kings is also a spade (the King of Spades), we need to account for this overlap when calculating the total.

**Hint:** The intersection refers to the cards that are counted in both categories (kings and spades).

### Step 5: Use the formula for the probability of the union of two events
The probability of drawing a king or a spade can be calculated using the formula:
\[ P(K \cup S) = P(K) + P(S) - P(K \cap S) \]
Where:
- \( P(K) \) is the probability of drawing a king.
- \( P(S) \) is the probability of drawing a spade.
- \( P(K \cap S) \) is the probability of drawing the King of Spades.

### Step 6: Calculate the probabilities
1. **Probability of drawing a king:**
\[ P(K) = \frac{4}{52} \]
2. **Probability of drawing a spade:**
\[ P(S) = \frac{13}{52} \]
3. **Probability of drawing the King of Spades:**
\[ P(K \cap S) = \frac{1}{52} \]

### Step 7: Substitute the values into the formula
Now substitute the values into the formula:
\[ P(K \cup S) = P(K) + P(S) - P(K \cap S) \]
\[ P(K \cup S) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} \]

### Step 8: Simplify the expression
Combine the fractions:
\[ P(K \cup S) = \frac{4 + 13 - 1}{52} = \frac{16}{52} \]

### Step 9: Reduce the fraction
Now, simplify \( \frac{16}{52} \):
\[ P(K \cup S) = \frac{4}{13} \]

### Final Answer
The probability that a card drawn at random from a pack of 52 cards is either a king or a spade is \( \frac{4}{13} \).

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