4 cards are drawn at random from a standard deck.
Find the probability that all the cards are hearts. 


Find the probability that all the cards are face cards. 

Note: Face cards are kings, queens, and jacks.

Find the probability that all the cards are even. 

(Consider aces to be 1, jacks to be 11, queens to be 12, and kings to be 13)

Let's break down each probability calculation separately.

### 1. **Probability that all 4 cards are hearts:**

A standard deck has 52 cards, and each suit (hearts, spades, diamonds, clubs) contains 13 cards. We need to find the probability that all 4 cards drawn are from the hearts suit.

- Total number of ways to choose 4 cards from the deck:  
  \[
  \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725
  \]

- Number of ways to choose 4 cards from the 13 hearts:  
  \[
  \binom{13}{4} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} = 715
  \]

- Probability that all 4 cards are hearts:  
  \[
  P(\text{4 hearts}) = \frac{\binom{13}{4}}{\binom{52}{4}} = \frac{715}{270725} \approx 0.00264
  \]

### 2. **Probability that all 4 cards are face cards (King, Queen, Jack):**

There are 12 face cards in the deck (3 face cards per suit and 4 suits), so we calculate the probability that all 4 cards drawn are face cards.

- Total number of ways to choose 4 cards from the deck:  
  \[
  \binom{52}{4} = 270725
  \]

- Number of ways to choose 4 face cards from the 12 face cards:  
  \[
  \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495
  \]

- Probability that all 4 cards are face cards:  
  \[
  P(\text{4 face cards}) = \frac{\binom{12}{4}}{\binom{52}{4}} = \frac{495}{270725} \approx 0.00183
  \]

### 3. **Probability that all 4 cards are even:**

The even-numbered cards in the deck are 2, 4, 6, 8, 10, and 12 (queen is considered 12). Each of these numbers appears in all 4 suits. Therefore, there are 6 even-numbered cards per suit and 4 suits, giving us 24 even-numbered cards in total.

- Total number of ways to choose 4 cards from the deck:  
  \[
  \binom{52}{4} = 270725
  \]

- Number of ways to choose 4 even-numbered cards from the 24 even cards:  
  \[
  \binom{24}{4} = \frac{24 \times 23 \times 22 \times 21}{4 \times 3 \times 2 \times 1} = 10626
  \]

- Probability that all 4 cards are even:  
  \[
  P(\text{4 even cards}) = \frac{\binom{24}{4}}{\binom{52}{4}} = \frac{10626}{270725} \approx 0.03926
  \]

---

### Summary of probabilities:
- Probability that all 4 cards are hearts: \( \approx 0.00264 \)
- Probability that all 4 cards are face cards: \( \approx 0.00183 \)
- Probability that all 4 cards are even: \( \approx 0.03926 \)

---

Would you like further details on any of these steps?

Here are 5 related questions to expand on this topic:

1. What is the probability that none of the 4 cards drawn are hearts?
2. What is the probability of drawing at least one face card?
3. What is the probability that all 4 cards are spades?
4. How does the probability change if 5 cards are drawn instead of 4?
5. What is the probability that all 4 cards are aces?

### Tip: 
When dealing with combinatorics problems, identifying the correct subset of items and ensuring you calculate the total and favorable outcomes properly is key to solving probability questions.

4 cards are drawn at random from a standard deck. Find the probability that all the cards are hearts. Find the probability that all the cards are face cards. Note: Face cards are kings, queens, and jacks. Find the probability that all the cards are even. (Consider aces to be 1, jacks to be 11, queens to be 12, and kings to be 13)

Explore the probability of drawing 4 hearts, face cards, or even-numbered cards from a deck. This problem covers key probability and combinatorics concepts, including the combination formula and basic probability theorems. Suitable for high school students (Grades 9-12).

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