# Rules

Tell me poker hands from lowest to the highest

In poker, hands are ranked from lowest to highest as follows:

1. **High Card** – No combination, just the highest card (e.g., Ace-high).
2. **One Pair** – Two cards of the same rank (e.g., 5♠ 5♦).
3. **Two Pair** – Two different pairs (e.g., 8♠ 8♦ 4♣ 4♠).
4. **Three of a Kind** – Three cards of the same rank (e.g., J♠ J♣ J♦).
5. **Straight** – Five consecutive cards of any suit (e.g., 6♠ 7♣ 8♦ 9♠ 10♥).
6. **Flush** – Five cards of the same suit, not in sequence (e.g., A♠ 10♠ 8♠ 5♠ 3♠).
7. **Full House** – Three of a kind plus a pair (e.g., K♠ K♣ K♦ 9♠ 9♣).
8. **Four of a Kind** – Four cards of the same rank (e.g., Q♠ Q♣ Q♦ Q♥).
9. **Straight Flush** – Five consecutive cards of the same suit (e.g., 5♠ 6♠ 7♠ 8♠ 9♠).
10. **Royal Flush** – The highest straight flush: A♠ K♠ Q♠ J♠ 10♠.

A Royal Flush is the best hand, while a High Card is the weakest.

***

Probability of Straight vs Straight Flush

The probabilities of getting a **Straight** and a **Straight Flush** in a standard 5-card poker hand (from a 52-card deck) are significantly different:

1. **Straight Probability**:
   * A straight consists of five consecutive ranks, but they can be of any suit.
   * There are **10 possible straights** (A-5, 2-6, ..., 10-A).
   * Each straight has **4⁵ (1,024) combinations** since each card in the straight can be one of 4 suits.
   * However, this count includes Straight Flushes, so we need to subtract them.
   * The probability of a straight (excluding straight flushes) is **4.62%** (or **1 in 21.7 hands**).
2. **Straight Flush Probability**:
   * A straight flush is a straight where all cards are of the same suit.
   * There are **10 possible straight flushes**, one for each consecutive sequence.
   * Each straight flush has exactly **4 ways** (one per suit).
   * This gives us **40 possible straight flush hands**.
   * The probability of a straight flush is **0.00139%** (or **1 in 72,193 hands**).

#### Comparison:

* A **Straight** is **about 3,330 times more common** than a **Straight Flush**.
* The **Royal Flush** (the highest straight flush) is even rarer, with a probability of **0.000154%** (or **1 in 649,740 hands**).

In summary, while straights are relatively frequent in poker, straight flushes are among the rarest hands you can get.

***

All hands probabilities

Here are the probabilities of all poker hands in a **5-card hand** from a standard **52-card deck**, ranked from most common to rarest:

| **Hand**            | **Combinations** | **Probability (%)** | **Odds (1 in X hands)** |
| ------------------- | ---------------- | ------------------- | ----------------------- |
| **High Card**       | 1,302,540        | 50.12%              | 1.99                    |
| **One Pair**        | 1,098,240        | 42.26%              | 2.37                    |
| **Two Pair**        | 123,552          | 4.75%               | 21.0                    |
| **Three of a Kind** | 54,912           | 2.11%               | 47.3                    |
| **Straight**        | 10,200           | 0.39%               | 255                     |
| **Flush**           | 5,108            | 0.20%               | 508                     |
| **Full House**      | 3,744            | 0.14%               | 693                     |
| **Four of a Kind**  | 624              | 0.024%              | 4,165                   |
| **Straight Flush**  | 40               | 0.00139%            | 72,193                  |
| **Royal Flush**     | 4                | 0.000154%           | 649,740                 |

#### Key Takeaways:

* **High Card and One Pair** dominate most hands.
* **Two Pair and Three of a Kind** occur much less often but still relatively frequently.
* **Straights and Flushes** are rare, while **Full House and Four of a Kind** are even harder to get.
* **Straight Flushes and Royal Flushes** are extremely rare, with a **Royal Flush appearing only once in 649,740 hands**.


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