A card is drawn at random from a pack of 52 playing cards. What is the probability that the card drawn is neither a spade nor a queen.

Answer

To find the probability that a card drawn from a pack of 52 playing cards is neither a spade nor a queen, we can follow these steps:

### Step 1: Determine the total number of cards
The total number of cards in a standard deck is 52.

### Step 2: Identify the number of spades
In a deck of cards, there are 13 spades.

### Step 3: Identify the number of queens
There are 4 queens in a deck (one from each suit: hearts, diamonds, clubs, and spades).

### Step 4: Count the overlap (spade queen)
Since one of the queens is also a spade, we need to ensure we do not double-count it. Therefore, we have:
- Total spades = 13
- Total queens = 4
- Overlapping card (queen of spades) = 1

### Step 5: Calculate the number of cards that are either spades or queens
Using the principle of inclusion-exclusion:
\[ \text{Number of spades or queens} = \text{Number of spades} + \text{Number of queens} - \text{Number of overlapping cards} \]
\[ = 13 + 4 - 1 = 16 \]

### Step 6: Calculate the number of cards that are neither spades nor queens
To find the number of cards that are neither spades nor queens, we subtract the number of spades or queens from the total number of cards:
\[ \text{Number of cards that are neither} = \text{Total cards} - \text{Number of spades or queens} \]
\[ = 52 - 16 = 36 \]

### Step 7: Calculate the probability
The probability that the card drawn is neither a spade nor a queen is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{neither spade nor queen}) = \frac{\text{Number of cards that are neither}}{\text{Total number of cards}} = \frac{36}{52} \]

### Step 8: Simplify the probability
To simplify \(\frac{36}{52}\), we can divide both the numerator and the denominator by their greatest common divisor, which is 4:
\[ \frac{36 \div 4}{52 \div 4} = \frac{9}{13} \]

### Final Answer
The probability that the card drawn is neither a spade nor a queen is \(\frac{9}{13}\).

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