The river: bet 1x or fold
Folding costs 2 antes for sure. Bet 1x when that's worse.
You've checked twice and all five community cards are out. Now it's bet 1x or fold.
Folding is easy to price: you lose your Ante and your Blind, exactly 2 antes, every time. So the question is: does betting 1x lose less than 2 antes on average?
If you bet 1x and win, you collect on Play and usually on the Ante (+2). If you lose, you lose Ante, Blind and Play (−3). You're only putting 1 extra Ante at risk, to protect the 2 you'd give up by folding. That's why the bar is low: when the dealer qualifies, winning about 1 time in 5 is enough.
Rule one is simple: with a hidden pair or better, always bet 1x. on (a pair of tens) has an EV of +0.261 for 1x against −2 for folding.
Without a pair it's closer, and you need to know how many dealer hands beat you. Lesson 12 shows a quick way to count. The "fewer than 21 outs" rule comes from the Wizard of Odds strategy, where the river rules are credited to James Grosjean. For now, look at the EV sheet:
| You | Board | Bet 1x | Fold | Best |
|---|---|---|---|---|
| +0.261 | −2 | 1x (hidden pair) | ||
| −1.821 | −2 | 1x (queen high, but enough wins) | ||
| −2.492 | −2 | Fold (eight high) | ||
| −0.136 | −2 | 1x (the board plays: mostly ties) |
loses money either way. Betting 1x loses 1.82 antes on average and folding loses 2, so you play. Your queen high beats every dealer hand that didn't pair, and when the dealer doesn't qualify your Ante can't lose.
When the board makes the best hand for everyone, as with , always bet 1x. Most of the time it's a tie and everything pushes, which is much better than losing 2 antes.
Show the math: The "1 in 5" bar
Ignore ties and Blind bonuses for a moment, and say the dealer qualifies. Bet 1x: win +2 (Ante + Play), lose −3 (Ante + Blind + Play). EV(1x) = 2w − 3(1 − w) = 5w − 3, which beats −2 when w > 1/5.
When the dealer doesn't qualify, the Ante pushes. Then a win pays +1 (Play) and a loss costs −2 (Blind + Play), so EV(1x) = w − 2(1 − w) = 3w − 2, which beats −2 for any w > 0. Against a dealer who missed, playing is never worse than folding. That's why "queen high" can be a 1x hand: many of its wins come against dealers who don't qualify.
Exact for on : of 990 dealer hands, you beat 256 (all non-qualifying), tie 9 and lose to 725 (609 qualifying, 116 not). EV(1x) = (256 × 1 − 609 × 3 − 116 × 2) / 990 = −601/330 ≈ −1.821.
Quiz
How much does folding on the river cost?
- 1 Ante
- 2 Antes
- 3 Antes
- It depends on the dealer's cards
Show the answer
B: 2 Antes. You give up your Ante and Blind: exactly 2 antes.
Quiz
You have a hidden pair on the river. What's the play?
- Always bet 1x
- Fold unless it's a high pair
- Bet 1x only if the board has no flush draw
- Fold
Show the answer
A: Always bet 1x. A hidden pair or better is always worth playing.
Quiz
Why can betting 1x be right even when its EV is negative (say −1.8 antes)?
- It can't
- Because folding is −2, which is worse
- Because the Blind might pay
- Because of the Trips bet
Show the answer
B: Because folding is −2, which is worse. Compare it with folding, not with zero.
Quiz
Board:
What's the best river play?
- Bet 1x
- Fold
Show the answer
A: Bet 1x. The board plays, so it's usually a tie. Betting 1x costs 0.136 antes on average, and folding costs 2.