A card is drawn from a well - shuffled pack of playing cards . What is the probability that it is elther a spade or an ace or both .

Answer

To find the probability that a card drawn from a well-shuffled pack of playing cards is either a spade or an ace or both, we can follow these steps:

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

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

### Step 2: Define the events
Let:
- Event A = the card drawn is a spade.
- Event B = the card drawn is an ace.

### Step 3: Calculate the probability of event A
In a deck, there are 13 spades.

\[ P(A) = \frac{\text{Number of spades}}{\text{Total number of cards}} = \frac{13}{52} \]

**Hint:** Simplifying fractions can help in calculations.

### Step 4: Calculate the probability of event B
In a deck, there are 4 aces (one in each suit: spade, heart, diamond, club).

\[ P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} \]

**Hint:** Ensure you count all suits when determining the number of aces.

### Step 5: Calculate the probability of both events occurring (intersection)
The intersection of A and B, denoted as \( P(A \cap B) \), is the probability that the card drawn is both a spade and an ace. There is only one ace of spades.

\[ P(A \cap B) = \frac{\text{Number of ace of spades}}{\text{Total number of cards}} = \frac{1}{52} \]

**Hint:** The intersection represents the overlap between the two events.

### Step 6: Use the formula for the union of two events
To find the probability that a card is either a spade or an ace or both, we use the formula:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 7: Substitute the values into the formula
Now we can substitute the probabilities we found:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
\[ P(A \cup B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} \]

### Step 8: Simplify the expression
Combine the fractions:

\[ P(A \cup B) = \frac{13 + 4 - 1}{52} = \frac{16}{52} \]

### Step 9: Simplify the final probability
Now simplify \( \frac{16}{52} \):

\[ P(A \cup B) = \frac{4}{13} \]

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

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