A card is drawn at random from a well shuffled pack of 52 cards. Find the probability that it is either an ace or spade.

Explanation

To find the probability that a card drawn from a well-shuffled pack of 52 cards is either an ace 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, each containing 13 cards.

### Step 2: Count the number of aces
In a deck, there are 4 aces (one for each suit: hearts, diamonds, clubs, and spades).

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

### Step 3: Count the number of spades
In a deck, there are 13 spades.

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

### Step 4: Identify the overlap (common cards)
One of the spades is also an ace (the ace of spades). This means we have counted it twice when we counted the aces and the spades.

**Hint:** Consider how many cards belong to both categories (aces and spades).

### Step 5: Use the formula for the probability of the union of two events
The probability of drawing either an ace or a spade can be calculated using the formula:
\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]
Where:
- \(P(E)\) is the probability of drawing an ace.
- \(P(F)\) is the probability of drawing a spade.
- \(P(E \cap F)\) is the probability of drawing a card that is both an ace and a spade.

### Step 6: Calculate the probabilities
1. Calculate \(P(E)\):
\[ P(E) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} \]

2. Calculate \(P(F)\):
\[ P(F) = \frac{\text{Number of spades}}{\text{Total number of cards}} = \frac{13}{52} \]

3. Calculate \(P(E \cap F)\):
\[ P(E \cap F) = \frac{\text{Number of aces that are spades}}{\text{Total number of cards}} = \frac{1}{52} \]

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

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

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

### Final Answer
The probability that the card drawn is either an ace or a spade is \(\frac{4}{13}\).

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