Find the probability that all the cards are face cards.
Note: Face cards are kings, queens, and jacks.

Question

Find the probability that all the cards are face cards. Note: Face cards are kings, queens, and jacks.

UpStudy AI · Accepted Answer

To determine the probability that all drawn cards are face cards (i.e., kings, queens, and jacks) from a standard deck of 52 playing cards, we'll follow these steps.

### Assumptions:
1. **Standard Deck:** The deck consists of 52 cards.
2. **Face Cards:** There are 12 face cards in total (4 Kings, 4 Queens, and 4 Jacks).
3. **Number of Drawn Cards:** We'll assume you're drawing **5 cards** without replacement (common in many card games like Poker).

### Step-by-Step Calculation:

1. **Total Number of Ways to Draw 5 Cards from 52:**
   
   The total number of possible 5-card combinations from a 52-card deck is given by the combination formula:
   
   $$
   \text{Total combinations} = \binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960
   $$

2. **Number of Ways to Draw 5 Face Cards from 12:**
   
   Similarly, the number of ways to draw 5 face cards from the 12 available face cards is:
   
   $$
   \text{Face card combinations} = \binom{12}{5} = \frac{12!}{5!(12-5)!} = 792
   $$

3. **Calculating the Probability:**
   
   The probability $$ P $$ that all 5 drawn cards are face cards is the ratio of the favorable outcomes to the total possible outcomes:
   
   $$
   P = \frac{\text{Face card combinations}}{\text{Total combinations}} = \frac{792}{2,598,960} \approx 0.000305
   $$
   
   To express this as a percentage:
   
   $$
   P \approx 0.0305\%
   $$

### Conclusion:

The probability of drawing **5 face cards** consecutively from a standard 52-card deck without replacement is approximately **0.0305%**.

**Note:** If you're drawing a different number of cards, you can adjust the numbers accordingly using the combination formula:

$$
P = \frac{\binom{12}{n}}{\binom{52}{n}}
$$

where $$ n $$ is the number of cards you are drawing.
The probability of drawing 5 face cards from a standard deck is about 0.03%.

Find the probability that all the cards are face cards.
Note: Face cards are kings, queens, and jacks.

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