In the card game of poker, a five card flush is any group of five cards that are all of the same suit (that is, all hearts, diamonds, clubs or spades). A flush is considered a "straight" flush when all five cards are also in sequence, and a straight flush is "royal" when it contains the ace, king, queen and jack. Using the basic rules of probability, you can easily calculate the probability of receiving any kind of flush when you or a dealer select five cards at random from a standard 52 card deck.
Probability of a Five Card Flush
Enter the number "1" into the calculator. This is the probability that the first card dealt will be any of the four suits. This probability, of course, is 100%, but it is normally expressed in the decimal form of 1 when calculating probability.
Multiply the number 1 just entered by 12 and divide by 51 for an answer of approximately 0.2353. The ratio 12/51 is the probability that the second card dealt is of the same suit as the first. This is because there are now 12 cards remaining of the same suit out of a total of 51 remaining cards in the deck; thus, the chance of picking one of those cards is 12 out of 51.
Multiply the number derived in the previous step by the following fractions: 11/50, 10/49 and 9/48. These fractions represent the probabilities of being dealt a card of the same suit on each of the subsequent three deals; with each subsequent deal, there is one less card of the suit and one less card of the overall deck, so both numbers of the fraction decrease by one. Multiplying the fractions gives an answer of approximately 0.00198. This number is the probability of being dealt a five card flush of any kind. In percentage terms, the probability is 0.198%.
Probability of a Straight Flush or Royal Flush
Calculate the total number of unique five card hands that you could pick or be dealt from a 52 card deck. This is calculated using the formula for combinations: C = 52!/[(5!)(52 - 5)!], where the the "!" sign means that the preceding number is to be multiplied by itself less 1, then less 2 and so on. The value of 5!, for example, would be 5 x (5-1) x (5 - 2) x ... x 1. The value of C derived from this calculation is 2,598,960 unique five card hands.
Determine the number of possible straight flush card hands that you could be dealt. You can calculate this by considering that there are 10 possible "straight" sequences; the straight beginning with the ace, with 2, with 3 and with all the other cards up to 10. Since these 10 possible sequences can be of any of the four suits, multiply the two numbers; since 10 x 4 = 40, there are 40 possible straight flush hands.
Determine the number of possible royal flush card hands. Since there is only one sequence of ace, king, queen, jack and ten in each suit, there are four possible royal flush hands.
Subtract the number of royal flush hands from the number of straight flush hands to find the number of straight flush hands that are not a royal flush. The answer is 36, since 40 - 4 = 36.
Divide the number of possible straight flush hands by the total possible number of five card hands. This equals 36/2,598,960 or 0.0000139. In percentage terms, the value is 0.00139%. This is the probability of receiving a straight flush that is not a royal flush.
Divide the number of possible royal flush hands by the total possible number of five card hands. The answer will be equal to 4/2,598,960 or 0.00000154. In percentage terms, the value is 0.000154%. This is the probability of being dealt a royal flush, and it is the lowest probability of any poker hand.