In poker, a royal flush is the highest-ranking hand and is extremely rare. The odds of hitting a royal flush can be calculated based on the number of possible combinations and the number of cards in a deck.

A royal flush consists of the following five cards in the same suit: Ace, King, Queen, Jack, and Ten. Since there are four suits in a deck (hearts, diamonds, clubs, and spades), the probability of getting a royal flush can be determined as follows:

1. Calculate the number of possible royal flush combinations:

– There are 4 possible Aces in a deck.

– For each Ace, there is only 1 King, 1 Queen, 1 Jack, and 1 Ten in the same suit.

– Therefore, there are 4 possible royal flush combinations.

2. Calculate the total number of possible 5-card combinations:

– In a standard deck, there are 52 cards.

– To calculate the number of 5-card combinations, we can use the formula for choosing k objects from a set of n objects without replacement: nCk = n! / (k!(n-k)!).

– In this case, n = 52 (total cards) and k = 5 (cards in a hand).

– Therefore, the total number of possible 5-card combinations is 52C5 = 2,598,960.

3. Calculate the probability of getting a royal flush:

– The probability is given by the number of favorable outcomes (4 royal flush combinations) divided by the total number of possible outcomes (2,598,960 combinations).

– Therefore, the probability is 4 / 2,598,960 ≈ 0.00000154 or approximately 0.000154%.

It’s worth noting that the odds of hitting a royal flush may vary depending on the specific poker variant being played and the number of cards dealt. The calculations above are based on the standard 5-card poker game.

In conclusion, the odds of hitting a royal flush in poker are extremely low, making it a highly prized and rare hand in the game. Players often consider it a jackpot hand due to its rarity and the excitement it brings to the table.