Two cards are drawn from a well shuffled pack of cards without replacement . Find the probability that neither a Jack nor a card of spades is drawn .

Answer

To solve the problem of finding the probability that neither a Jack nor a card of Spades is drawn when two cards are drawn from a well-shuffled pack of 52 cards without replacement, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total number of cards**:
A standard deck has 52 cards.

2. **Determine the number of Spades and Jacks**:
- There are 13 Spades in the deck.
- There are 4 Jacks in total (one from each suit).

3. **Calculate the total number of non-Spade cards**:
\[ \text{Total non-Spade cards} = 52 - 13 = 39 \]

4. **Determine the number of Jacks that are not Spades**:
Since there is 1 Jack of Spades, the number of Jacks that are not Spades is:
\[ \text{Non-Spade Jacks} = 4 - 1 = 3 \]

5. **Calculate the total number of cards that are neither Jacks nor Spades**:
\[ \text{Total cards that are neither Jacks nor Spades} = 39 - 3 = 36 \]

6. **Calculate the total number of ways to draw 2 cards from the deck**:
The total number of ways to choose 2 cards from 52 is given by the combination formula:
\[ \text{Total ways to choose 2 cards from 52} = \binom{52}{2} = \frac{52 \times 51}{2} = 1326 \]

7. **Calculate the number of ways to draw 2 cards that are neither Jacks nor Spades**:
The total number of ways to choose 2 cards from the 36 cards that are neither Jacks nor Spades is:
\[ \text{Ways to choose 2 cards from 36} = \binom{36}{2} = \frac{36 \times 35}{2} = 630 \]

8. **Calculate the probability**:
The probability that neither a Jack nor a Spade is drawn is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{neither Jack nor Spade}) = \frac{\text{Ways to choose 2 cards from 36}}{\text{Total ways to choose 2 cards from 52}} = \frac{630}{1326} \]

9. **Simplify the fraction**:
To simplify \( \frac{630}{1326} \):
- The GCD of 630 and 1326 is 126.
- Thus, \( \frac{630 \div 126}{1326 \div 126} = \frac{5}{11} \).

### Final Answer:
The probability that neither a Jack nor a card of Spades is drawn is \( \frac{5}{11} \).