A card is drawn from a well shuffled pack of 52 cards. What is the probability that neither a spade nor a king will drawn?

Explanation

To find the probability that neither a spade nor a king will be drawn from a well-shuffled pack of 52 cards, 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: Count the number of spades
In a deck of cards, there are 13 spades.

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

### Step 4: Identify the overlap (spade that is also a king)
Since one of the kings is also a spade, we need to account for this overlap. Therefore, there is 1 card that is both a spade and a king.

### Step 5: Calculate the probability of drawing a spade or a king
Using the principle of inclusion-exclusion, the probability of drawing a spade or a king can be calculated as follows:

\[ P(\text{Spade or King}) = P(\text{Spade}) + P(\text{King}) - P(\text{Spade and King}) \]

Calculating each probability:
- Probability of drawing a spade:
\[ P(\text{Spade}) = \frac{13}{52} \]

- Probability of drawing a king:
\[ P(\text{King}) = \frac{4}{52} \]

- Probability of drawing a card that is both a spade and a king:
\[ P(\text{Spade and King}) = \frac{1}{52} \]

Now substituting these values into the inclusion-exclusion formula:

\[ P(\text{Spade or King}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} \]

### Step 6: Simplify the probability of drawing a spade or a king
\[ P(\text{Spade or King}) = \frac{16}{52} = \frac{4}{13} \]

### Step 7: Calculate the probability of drawing neither a spade nor a king
The probability of drawing neither a spade nor a king is given by:

\[ P(\text{Neither Spade nor King}) = 1 - P(\text{Spade or King}) \]

Substituting the value we found:

\[ P(\text{Neither Spade nor King}) = 1 - \frac{4}{13} = \frac{13 - 4}{13} = \frac{9}{13} \]

### Final Answer
Thus, the probability that neither a spade nor a king will be drawn is:

\[ \frac{9}{13} \]

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