Counting the dealer's outs

Lesson 12 of 23 · Perfect strategy I: lowest edge · 9 min read

Fewer than 21 cards that beat you: bet 1x.

Without a hidden pair, you need a quick way to tell "queen high, play it" from "eight high, fold it". The trick is to imagine one card in the dealer's hand and ask: which cards would beat me? Those are the dealer's outs. This counting method follows the one in the Wizard of Odds strategy¹, credited there to James Grosjean². You can see 7 cards, which leaves 45 unseen.

How to count, step by step:

1. Pairs with the board. Every board card you don't pair gives the dealer 3 outs (the three other cards of that rank). Five unpaired board ranks = 15 outs.
2. Higher kickers. Every rank that beats your best card, and isn't on the board, adds 4 outs. A card you hold counts as seen.
3. Straights and flushes. Add any single card that would complete one for the dealer.

Then: fewer than 21 outs → bet 1x. 21 or more → fold.

Example 1. Q♣︎7♦︎ on K♦︎J♥︎9♠︎5♣︎2♦︎. Board pairs: K, J, 9, 5, 2 = 15. Higher than your queen and not on the board: only the ace = 4. No straight or flush is one card away. 19 outs → bet 1x.

Example 2. 8♣︎6♦︎ on K♥︎Q♠︎4♦︎3♣︎2♥︎. Board pairs = 15. Cards that beat your K-Q-8 kicker: aces, jacks, tens and nines = 16. 31 outs → fold.

Example 3. J♦︎8♣︎ on the same board K♥︎Q♠︎4♦︎3♣︎2♥︎. Board pairs = 15, and only aces beat your K-Q-J = 4. 19 outs → bet 1x. One card made the difference.

Example 4. J♣︎9♦︎ on 8♠︎8♥︎5♣︎3♦︎2♠︎. The board is paired, so you're playing 8-8 with J-9-5. The dealer beats you with an ace, king or queen kicker (12), a 5, 3 or 2 for two pair (9), or the last two eights (2). 23 outs → fold.

How good is the rule? Very good. In a random sample of 3,000 river spots, it disagreed with perfect play on about 1 in 90 (≈ sampled). Nearly all of those are at 19 or 20 outs, where perfect play folds by a few hundredths of an Ante. One example is 2♣︎4♥︎ on Q♣︎Q♥︎K♥︎Q♠︎9♦︎: 19 outs, but folding is better (1x is −2.124). Following the rule instead of perfect river play costs 0.11% of the Ante, about 33¢ an hour at $10.

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