The calculation of probabilities begins with the computation of all possible hands. You can look up the formula online but the result is there are about 2.6 million possible hands. Every one of the hands has the same likelihood to come up during play.
The next step is to remove the No Pair hands including Straight Flushes, Nothings, Straights, Flushes, and Royal Straight Flushes. This means we are looking only at hands that have at a minimum one pair.
Without pairs the cards in your hand will have five different denominations randomly dealt from the thirteen existing cards in each suit. All five categories will select one set from the four available sets.
The Straight Flush comprises of five cards in a row in the same set and may have a high card of Ace, King, Queen, Jack, 10, 9,8,7,6, or 5 for ten different ranks. This means there are forty potential Straight Flushes. An Ace high Straight Flush is a Royal Flush. Subtract the four Royal Straight Flushes from the previous result and you have thirty-six other Straight Flushes which are either King high or lesser. This is 36 or 40 out of almost 2.6 million possible hands.
A Flush comprises of any five of the thirteen cards from one suit of the four available suits or for a total of 5148 possible flushes including the straight flushes. That means your chances for flush are 5108 over 2,598,960 or roughly 1 in every 500 hands.
A Straight comprises of five cards with successive denominations and can have a high card of Ace, King, Queen, Jack, 10, 9,8,7,6, or 5 for a total of ten different ranks. This includes again the forty potential Straight Flushes. Subtract forty, which leaves 10,200 potential regular Straights, or roughly 1 in every 250 hands.
“Nothing” hands are what is left once all the valuable hands are taken out. When you do the math it comes out to be a little over one in every two hands is worth nothing. And that is where the players skills become so important.