A poker hand consists of five cards randomly dealt from a standard deck of 52 cards. The order of the cards does not matter. Determine the following probabilities for a 5-card poker hand.

Write your answers in percent form, rounded to 4 decimal places, and don't forget to include a percent sign, %, in your answer..

Determine the probability that exactly 4 of these cards are Aces.
Answer: 
Determine the probability that all five of these cards are Spades.
Answer: 
Determine the probability that exactly 4 of these cards are face cards.
Answer: 
Determine the probability of selecting exactly 2 Aces and exactly 2 Kings
Answer: 
Determine the probability of selecting exactly 1 Jack.
Answer: 


To solve these probability problems, we use combinatorics and the concept of probability based on drawing from a deck of 52 cards. 

1. **Determine the probability that exactly 4 of these cards are Aces.**

   - Total possible 5-card hands: 
     \[
     \binom{52}{5} = \frac{52!}{5!(52 - 5)!} = 2,598,960
     \]
   - To have exactly 4 Aces in a hand, we select 4 Aces from the 4 available Aces and one non-Ace from the remaining 48 cards:
     \[
     \binom{4}{4} \times \binom{48}{1} = 1 \times 48 = 48
     \]
   - Probability:
     \[
     \frac{48}{2,598,960} \approx 0.0018\% 
     \]

   **Answer**: \(0.0018\%\)

2. **Determine the probability that all five of these cards are Spades.**

   - There are 13 Spades in the deck. We need to draw all 5 cards from these 13.
   - Number of ways to select 5 Spades from 13:
     \[
     \binom{13}{5} = \frac{13!}{5!(13 - 5)!} = 1,287
     \]
   - Probability:
     \[
     \frac{1,287}{2,598,960} \approx 0.0495\%
     \]

   **Answer**: \(0.0495\%\)

3. **Determine the probability that exactly 4 of these cards are face cards.**

   - Face cards in a deck are Jacks, Queens, and Kings, totaling 12 face cards (3 face cards per suit).
   - To have exactly 4 face cards, select 4 from the 12 face cards and 1 non-face card from the remaining 40 cards:
     \[
     \binom{12}{4} \times \binom{40}{1} = 4,950 \times 40 = 198,000
     \]
   - Probability:
     \[
     \frac{198,000}{2,598,960} \approx 7.6180\%
     \]

   **Answer**: \(7.6180\%\)

4. **Determine the probability of selecting exactly 2 Aces and exactly 2 Kings.**

   - To have exactly 2 Aces and 2 Kings, we select 2 from the 4 available Aces, 2 from the 4 Kings, and 1 card from the remaining 44 cards:
     \[
     \binom{4}{2} \times \binom{4}{2} \times \binom{44}{1} = 6 \times 6 \times 44 = 1,584
     \]
   - Probability:
     \[
     \frac{1,584}{2,598,960} \approx 0.0610\%
     \]

   **Answer**: \(0.0610\%\)

5. **Determine the probability of selecting exactly 1 Jack.**

   - We select exactly 1 Jack from the 4 available Jacks, and the remaining 4 cards from the 48 non-Jacks:
     \[
     \binom{4}{1} \times \binom{48}{4} = 4 \times 194,580 = 778,320
     \]
   - Probability:
     \[
     \frac{778,320}{2,598,960} \approx 29.9524\%
     \]

   **Answer**: \(29.9524\%\)

---

Would you like a breakdown of any specific steps or further clarification? 

### Related Questions

1. What is the probability of drawing a hand with at least one Ace?
2. How would you calculate the probability of drawing three of a kind in a poker hand?
3. What is the probability of drawing a hand that contains no face cards?
4. How do probabilities differ between 5-card and 7-card hands in poker?
5. Can the probability of a full house be calculated similarly?

**Tip**: When dealing with "exactly" conditions, remember to consider only the specific card combinations that match the conditions and to ignore any unwanted combinations.

A poker hand consists of five cards randomly dealt from a standard deck of 52 cards. The order of the cards does not matter. Determine the following probabilities for a 5-card poker hand. Write your answers in percent form, rounded to 4 decimal places, and don't forget to include a percent sign, %, in your answer.. Determine the probability that exactly 4 of these cards are Aces. Answer: Determine the probability that all five of these cards are Spades. Answer: Determine the probability that exactly 4 of these cards are face cards. Answer: Determine the probability of selecting exactly 2 Aces and exactly 2 Kings Answer: Determine the probability of selecting exactly 1 Jack. Answer:

Explore probability calculations for specific 5-card poker hands, including the chance of drawing exactly 4 Aces, 5 Spades, 4 face cards, and more. Detailed solutions cover combinatorial methods and probability formulas, ideal for advanced high school students studying probability and combinatorics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation