How to Calculate the Odds of Getting a Royal Flush
A royal flush is when you have the 10, jack, queen, king, and ace of a single suit. To calculate the odds of getting a royal flush on a single deal from a deck of 52 cards involves a mathematical function called the factorial (represented by an exclamation point). For a positive integer, this is simply multiplying the number by all the lesser positive integers down to 1. For example, 4! is 4x3x2x1=24.
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Instructions
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Calculate the total number of possible five card hands. The possible number of five card hands is calculated using the function n!/[(n-k)! x k!], where n is 52 and k is 5. This becomes 52!/(47! x 5!) = 2,598,960 possible distinct five-card hands.
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Create a ratio of possible royal flushes to total possible outcomes. There are four possible royal flushes, one for each suit. As calculated above, there are 2,598,960 possible five-card hands. This means the ratio of royal flushes to possible five-card hands is 4:2,598,960.
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Calculate the probability of getting a royal flush. Probability is simply the first term of the ratio divided by the second. In this case, 4/2,598,960 is approximately 0.00000154.
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Convert probability to odds. To convert probability (P) into odds use the expression (1/P)-1:1. Substituting the probability calculated in the previous step (0.00000154) for P yields (1/0.00000154)-1:1. The quantity in parentheses equals approximately 649,740. Therefore the final odds on drawing a royal flush is 649,739:1, or one out of every 649,740 draws.
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Tips & Warnings
In a actual game setting, you can calculate the odds of hitting a royal flush by first finding the number of "outs" or useful cards remaining in the deck, and multiplying a ratio of the number "bad" cards to unknown cards for each remaining draw.
In Texas Hold 'Em, after the flop you will need somewhere between zero and five cards to get a royal flush. The ratio for an unsuccessful draw is the difference of 47 minus this number, divided by 47. This value should be multiplied by the ratio for an unsuccessful draw on the river (46 minus the number of remaining outs, divided by 46). To convert this value into a percentage, subtract it from one and multiply this difference by 100. For example, if the product is 0.76, indicating a 76 percent chance of drawing an unsuccessful card, the odds of drawing a successful card is 1 - 0.76 = 0.24, or 24 percent.
