Red queens and black jacks are removed from a pack of 52 playing cards. Find the probability that the card drawn from the remaining cards is:
neither a jack nor a king

Answer

To solve the problem of finding the probability that the card drawn from the remaining cards is neither a jack nor a king, we can follow these steps:

### Step 1: Identify the total number of cards after removal
We start with a standard deck of 52 playing cards. We are told that 2 red queens and 2 black jacks are removed from the deck.

- **Total cards initially**: 52
- **Cards removed**: 2 red queens + 2 black jacks = 4 cards

**Remaining cards**:
\[ 52 - 4 = 48 \]

### Step 2: Determine the number of jacks and kings remaining
Next, we need to find out how many jacks and kings are left in the remaining cards.

- **Jacks removed**: 2 (black jacks)
- **Remaining jacks**:
\[ 4 - 2 = 2 \]
- **Kings removed**: 0 (no kings were removed)
- **Remaining kings**:
\[ 4 \]

### Step 3: Calculate the probability of drawing a jack or a king
We need to find the probability of drawing either a jack or a king (denoted as \( P(J \cup K) \)).

Using the formula for the probability of the union of two events:
\[ P(J \cup K) = P(J) + P(K) - P(J \cap K) \]

Since a jack and a king cannot be the same card, \( P(J \cap K) = 0 \).

Now we calculate \( P(J) \) and \( P(K) \):
- **Probability of drawing a jack**:
\[ P(J) = \frac{\text{Number of remaining jacks}}{\text{Total remaining cards}} = \frac{2}{48} \]
- **Probability of drawing a king**:
\[ P(K) = \frac{\text{Number of remaining kings}}{\text{Total remaining cards}} = \frac{4}{48} \]

Now substituting these values into the union formula:
\[ P(J \cup K) = \frac{2}{48} + \frac{4}{48} - 0 = \frac{6}{48} = \frac{1}{8} \]

### Step 4: Calculate the probability of drawing neither a jack nor a king
To find the probability of drawing neither a jack nor a king (denoted as \( P(J' \cap K') \)), we use the complement rule:
\[ P(J' \cap K') = 1 - P(J \cup K) \]
Substituting the value we found:
\[ P(J' \cap K') = 1 - \frac{1}{8} = \frac{7}{8} \]

### Final Answer
The probability that the card drawn from the remaining cards is neither a jack nor a king is:
\[ \frac{7}{8} \]

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