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Part 1
In​ 5-card poker, find the probability of being dealt the following hand. Refer to the table. Note that a standard deck of playing cards has 52 cardslong dash—4 suits​ (clubs, diamonds,​ hearts, spades), where each suit has 13 cards​ (Ace, 2,​ 3, 4,​ 5, 6,​ 7, 8,​ 9, 10,​ Jack, Queen,​ King).
a diamonds flusha diamonds flush ​(not a royal flush or a straight​ flush)
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Part 1
The probability of being dealt a diamonds flusha diamonds flush is enter your response here.
​(Type an integer or decimal rounded to eight decimal places as​ needed.)

To calculate the probability of being dealt a flush in diamonds (not a royal flush or a straight flush) in a 5-card poker hand, we can follow these steps:

### Total possible hands in poker:
The total number of possible 5-card hands from a deck of 52 cards is computed using the combination formula \( C(n, r) \), which gives the number of ways to choose \( r \) objects from \( n \) without regard to the order:

\[
C(52, 5) = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!}
\]

This gives the total number of possible hands in a 52-card deck.

### Number of ways to get a diamonds flush (non-royal, non-straight):
To get a flush in diamonds, we must select 5 cards all from the 13 diamonds. The number of ways to choose 5 cards from the 13 diamonds is:

\[
C(13, 5) = \frac{13!}{5!(13-5)!} = \frac{13!}{5!8!}
\]

However, we must exclude any royal flush (Ace, King, Queen, Jack, 10) and any straight flush (five consecutive cards of the same suit). There is only **1 royal flush** (Ace, King, Queen, Jack, 10 of diamonds) and **9 possible straight flushes** (e.g., Ace to 5, 2 to 6, etc.).

Thus, the number of flushes without a straight or royal flush is:

\[
C(13, 5) - 10 = \text{valid flushes}
\]

### Probability calculation:
Finally, the probability is:

\[
P(\text{flush}) = \frac{\text{valid flush hands}}{C(52, 5)}
\]

Let me calculate the exact value for you.The probability of being dealt a diamonds flush (not a royal flush or a straight flush) in a 5-card poker hand is approximately \( 0.00049135 \) (rounded to eight decimal places).

Do you want further details or have any questions?

Here are some related questions for further exploration:
1. What is the probability of being dealt a royal flush in diamonds?
2. How does the probability of getting a flush change for a different suit?
3. What is the probability of being dealt a straight flush in any suit?
4. How do the odds of getting a flush compare with the odds of getting a full house?
5. What is the probability of getting a four-of-a-kind hand in poker?

**Tip:** In poker, the odds of specific hands depend heavily on the combinations of cards available, so understanding how to calculate combinations is key!

In 5-card poker, find the probability of being dealt the following hand. Refer to the table. Note that a standard deck of playing cards has 52 cards—4 suits (clubs, diamonds, hearts, spades), where each suit has 13 cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King). A diamonds flush (not a royal flush or a straight flush). The probability of being dealt a diamonds flush is (Type an integer or decimal rounded to eight decimal places as needed).

Learn how to calculate the probability of being dealt a diamonds flush in a 5-card poker hand, excluding royal and straight flushes. This detailed explanation covers probability and combinatorics, using key formulas such as combinations and basic probability. Suitable for Grades 11-12 students.

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