## introduction

Draw poker is played with a standard deck of 52 cards: 13 ranks (Ace through 10, and jack, queen, king) in 4 suits. (An ace usually counts as higher than a king, unless it is being used as “1” in a run.) Each player is dealt 5 cards.

Before I ever played poker – at home – I was required to learn a mantra: one pair, two pair, three of a kind… straight, flush, full house… four of a kind, straight flush. From least to best, those are the possible winning hands. (In fact, I will distinguish a royal straight flush from other straight flushes.)

What is the probability of being dealt each one of them?

That’s the first question I’ll answer, in several parts.

After the cards have been dealt and bet on, each player may discard some and get replacements. In a second post I will ask and answer: what are the odds against improving any given hand?

There is an excellent wiki article out there. In fact, the nice way to calculate the probabilities comes from it (especially for one pair, and for nothing).

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