As a seasoned expert in the field of probability and card games, specifically poker, I can provide a detailed analysis of the odds of getting a flush in a standard 52-card deck. A flush in poker is a hand where all five cards are of the same suit but not in a sequence, which is different from a straight flush where the cards are not only of the same suit but also in a sequence.

To calculate the odds of getting a flush, we first need to understand the total number of possible five-card hands that can be dealt from a standard deck. This is calculated by the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. In this case, \( n = 52 \) and \( k = 5 \), so the total number of possible hands is \( C(52, 5) \).

Now, to find the number of ways to get a flush, we need to consider that there are four suits in a deck, and for each suit, we can choose any five cards. Since the order of the cards does not matter, we use the combination formula again for each suit, which is \( C(13, 5) \), because there are 13 cards of each suit. Since there are four suits, we multiply this by four.

The probability of getting a flush is the number of flushes divided by the total number of possible hands. The odds, on the other hand, are the inverse of the probability, or the ratio of the number of hands that are not flushes to the number of flushes.

Let's do the math:

- Total number of possible hands: \( C(52, 5) = \frac{52!}{5!(52-5)!} \)
- Number of flushes: \( 4 \times C(13, 5) = 4 \times \frac{13!}{5!(13-5)!} \)

The odds of getting a flush are then calculated as:

\[ \text{Odds of flush} = \frac{C(52, 5) - 4 \times C(13, 5)}{4 \times C(13, 5)} \]

After calculating these values, we can express the odds in a more understandable format, such as "to 1" odds.

Now, regarding the royal flush mentioned, which is an Ace, King, Queen, Jack, and 10 all of the same suit, it is indeed a type of straight flush. However, the odds for a royal flush are different from a regular flush because there are only four possible royal flushes (one for each suit), making the odds significantly lower.

The odds of getting a royal flush are calculated similarly but with a much smaller numerator, as there is only one combination per suit. The probability is \( \frac{4}{C(52, 5)} \), and the odds are \( \frac{C(52, 5)}{4} \), which gives us the 649,739 to 1 odds mentioned.

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The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.read more >>