Pat hands: how often five cards need no exchange

In draw poker the first question about a hand is which cards to keep. For a small group of hands the answer is all of them. Those are called pat hands, and the exact count of them is a short sum from the standard table of five-card hands.

What "pat" means

A pat hand is one in which all five cards take part in the made hand, so exchanging any card can only make it worse or leave it unchanged. In practice that means a straight or better: a straight, a flush, a full house, four of a kind or a straight flush (the royal flush is the highest straight flush). Three of a kind, two pair and one pair are not pat, because they leave two, one or three cards doing nothing. Those spare cards are the ones you would normally exchange.

The word is used loosely. Some players call any hand they will not break "pat", and in some games a particular straight or flush is worth breaking for a stronger draw. This post uses the plain definition: straight or better, nothing to exchange.

Counting the pat hands

A standard 52-card deck deals C(52, 5) = 2,598,960 different five-card hands, all equally likely. Each pat category has a short count:

• Straight flush (40): ten possible starting ranks, from ace-low up to ace-high, times four suits. 36 of them are ordinary straight flushes and 4 are royal flushes.
• Four of a kind (624): 13 ranks for the four cards, then any of the remaining 48 cards: 13 × 48.
• Full house (3,744): 13 ranks and 4 ways to pick three of them, then 12 ranks and 6 ways to pick the pair: 13 × 4 × 12 × 6.
• Flush (5,108): four suits times C(13, 5) = 1,287 rank sets is 5,148, minus the 40 that are straight flushes.
• Straight (10,200): ten starting ranks, each card in one of four suits, is 10 × 4⁵ = 10,240, minus the same 40 straight flushes.

Add them up: 10,200 + 5,108 + 3,744 + 624 + 36 + 4 = 19,716. Divide by 2,598,960 and the share is 0.76% (0.759% to three places). Put the other way round, one deal in about 132 is pat, since 2,598,960 ÷ 19,716 is 131.8. The other 2,579,244 hands, 99.24% of all deals, have at least one card you could reasonably exchange.

Where the 19,716 come from

The mix is lopsided. Straights and flushes are 15,308 of the 19,716, which is 77.6%. The rest is thin:

Straights alone outnumber the other four categories combined (10,200 against 9,516). A full house or better is just 4,408 hands, or 0.17%: about one deal in 590. The ordering explains why a flush beats a straight, which Why a flush beats a straight covers in full.

The rest of the deals

Split every deal by what you would do with it and three groups appear:

1. High card, 1,302,540 hands (50.12%). No pair, no straight, no flush. You are looking for a draw or starting from scratch.
2. Pair, two pair or three of a kind, 1,276,704 hands (49.12%). Made hands with spare cards. These are the ordinary exchange decisions.
3. Pat, 19,716 hands (0.76%). Nothing to exchange.

The three counts add to 2,598,960. For the middle group, The odds of improving a pair when you exchange cards works through what the exchange is worth.

How long you wait

A chance of 1 in 132 does not mean one every 132 hands. Treat each deal as independent, with the pat chance p = 19,716 ÷ 2,598,960 = 0.759%, and the chance of at least one pat hand in n deals is 1 − (1 − p)ⁿ:

• In 50 deals: 31.7%.
• In 100 deals: 53.3%.
• In 132 deals: 63.4%, not 100%.
• In 500 deals: 97.8%.

So an evening of 100 hands is only slightly better than a coin toss to include a single pat hand, and two or three in a short spell is as ordinary as none.

Where this meets Poker vs Dealer

In Poker vs Dealer: Offline your five cards compete against the dealer and for the table pot. The store listing says you can hold your hand, pay an ante to forge a card by exchanging it for a fresh one, or bail and recover half of your committed chips. A pat hand is the clearest case for holding, because there is no card worth paying to replace. The listing also says the game uses its own five-card rules and has an in-game rules and payouts screen, so check there for how hands are ranked and paid. The counts above are standard five-card maths for a dealt hand, not a statement about how often the app deals them.

The useful takeaway is proportion. With about one deal in 132 pat, nearly every hand is a decision, and the practice worth getting is in the other 99.24%.