What is Zeebo's theorem in poker?

The saying that no player can fold a full house, on any street and for any bet. It is a rule of thumb about how people play, named after the online player Captain Zeebo. It has no proof, and players do sometimes fold one.

Captain Zeebo, also written captZEEbo, was the screen name of a high-stakes online player of the mid-2000s. ThePokerBank and BetMGM name him as Greg Lavery, and they date the theorem to a blog post of February 12, 2006, which survives in the Internet Archive.

How often does a full house lose?

Against one random hand at showdown it wins 86.99%, ties 7.17% and loses 5.84% of the time, an exact count of every deal. Against three random hands it loses 15.5% and against eight 31.7%, simulated. On a board that is not a full house itself, against an opponent who also holds a full house or better, it wins 33.86% and loses 42.98%.

Should you ever fold a full house?

Rarely, and only with a strong read. It beats or ties 94.16% of random hands, so a fold needs real evidence that the other player holds a bigger one. The case is best for an underfull on a board where a bigger one is easy to make, against a player who rarely bluffs. Against a full house or better an underfull wins 28.32% and loses 48.43%.

Is a full house the same as a boat?

Yes. A boat is another name for a full house: three cards of one rank and two of another.

Does Zeebo's theorem apply to other hands?

Not as stated: it names only the full house, and the figures here are for full houses alone.

Zeebo's Theorem: How Often a Full House Really Loses

Q: What beats a full house?

A bigger full house, four of a kind or a straight flush, which includes the royal flush. When a full house loses to one random hand, 76.84% of the time it is to a bigger full house.

What Zeebo's theorem says, and where it came from

In a blog post of February 12, 2006, under the heading The Full House Theorem, Captain Zeebo (also written captZEEbo) wrote: "Nobody is capable of folding a full house on any street for any sized bet, ever." He meant small full houses as well as big ones: "I'm not even referring to NUT full houses". The blog has gone, but the Internet Archive kept the post. The version you will usually see is a rewording that ThePokerBank and BetMGM print: "No player is capable of folding a full house on any betting round, regardless of the size of the bet." Both trace it to that post. Captain Zeebo was the screen name of a high-stakes online player they name as Greg Lavery. PokerTerms credits the phrase to the Two Plus Two forum poster CaptZEEbo, and PokerNews files it under the forum's classic days in a shorter form: nobody folds a full house. So the sources agree on the player and the era, and not on whether it first appeared on his blog or on the forum. The blog post is the earliest dated text we found. It is not a theorem in the mathematical sense, since nothing is proved. It is a claim about what people do, made with a straight face.

Is Zeebo's theorem true? How often a full house loses

Cards can test one part of it: is a full house strong enough that folding is nearly always wrong? You end with a full house on 2.60% of deals, about 1 in 38.5, the 3,473,184 of 133,784,560 seven-card hands on the hand probabilities page. Here is every one of those deals at showdown against random hands that never fold:

A river full house at showdown, by number of opponents

Opponents	Wins	Ties	Loses	Method
1	86.99%	7.17%	5.84%	Exact
2	80.16%	8.88%	10.96%	Exact
3	74.4%	10.2%	15.5%	Simulated
4	69.4%	11.1%	19.5%	Simulated
5	65.1%	11.8%	23.0%	Simulated
6	61.4%	12.3%	26.2%	Simulated
7	58.2%	12.7%	29.1%	Simulated
8	55.3%	13.0%	31.7%	Simulated

Every (hand, board) deal there is was visited once: 2,598,960 boards with 1,081 holdings each, valued with the site's hand evaluator. Against one hand, each full house was compared with all 990 holdings that share no card with it, and against two with every pair of those that share no card with each other. Those figures are exact counts. Against three or more an exact count is out of reach, so each of the 72,936,864 full houses met one seeded random set of opponents; no simulated figure has a standard error above 0.01 points. These are win rates, not equity: "wins" means alone, and a tie is shown separately. The good hand guide has the same figures for every other hand.

The boards that endanger a full house

Every full house in Hold'em needs a paired board. A full house is five cards of two ranks and at most two can come from your hand, so at least three board cards share two ranks. On a board with one pair it takes both your hole cards, few other holdings can do the same, and it loses to one random hand only 0.83% of the time. With three of a kind on the board it is another matter: any pocket pair makes a full house, the last card of the rank makes four of a kind, and a full house there loses to one random hand 14.66% of the time and to eight 68.3%:

Full houses by what the board holds, and how often they lose

The board holds	Share of full houses	Loses to 1	Loses to 3	Loses to 8
One pair	40.65%	0.83%	2.5%	6.5%
Two pair	30.32%	4.71%	13.4%	31.2%
Three of a kind	23.72%	14.66%	37.0%	68.3%
A full house	5.31%	11.31%	30.7%	64.4%

The first two columns are exact and the last two simulated, with a standard error under 0.03 points. The paired board odds show how often the board pairs at all.

Which full houses lose: top boat, underfull and playing the board

Full house and boat mean the same thing, and on one board there can be several sizes of it. The top full house is the biggest one anyone can make on that board; only four of a kind or a straight flush can beat it. An underfull has its three of a kind below its pair, like 8-8-8-K-K, and loses to any bigger full house on the same board. Playing the board means the five board cards are your hand and your hole cards add nothing. The last two columns are the ones that should worry you. They suppose the other player holds a full house or better, which is closer to what a big raise on a paired board means:

Each kind of full house, against one random hand and against a full house or better

Full house	Share	Wins vs random	Ties vs random	Loses vs random	Wins vs boat or better	Loses vs boat or better
Top full house	11.90%	97.54%	2.12%	0.34%	72.25%	3.02%
Lower, three of a kind over the pair	35.89%	90.36%	3.27%	6.36%	33.08%	44.15%
Underfull, three of a kind under the pair	47.30%	90.80%	2.98%	6.22%	28.32%	48.43%
Playing the board	4.91%	0%	88.31%	11.69%	0%	11.69%

All exact. "Boat or better" counts only the opponent's holdings that are a full house, four of a kind or a straight flush on that board, each with its true weight. A board that is a full house itself is left out of those two columns, except in the last row, because every holding on it is a boat and that says nothing about a raise. Across the other boards the record is 33.86% won, 23.16% tied and 42.98% lost. By rank, aces full loses to one random hand 4.02% of the time and deuces full 7.70%.

What beats a full house, and how often each one shows up

Three hands beat a full house: a bigger full house, four of a kind and a straight flush, royal flushes included. Against one random hand 76.84% of a full house's losses are to a bigger full house, 22.92% to four of a kind and 0.23% to a straight flush. So when a full house loses it is usually to another full house. 2.29% of full houses cannot be beaten by any two cards at all, and 44.45% are beaten by at most ten of the 990 holdings:

What a full house loses to

Beaten by	Share of losses to 1	To 3	To 8
A bigger full house	76.84%	73.8%	65.9%
Four of a kind	22.92%	25.9%	33.7%
A straight flush	0.23%	0.3%	0.3%

The first column is exact. The others are simulated: the best hand against the full house on each deal it lost, with a standard error under 0.02 points. The hand rankings show the full order of hands.

The full house that was folded

The theorem has been broken on camera. At the 2019 PokerStars Players Championship in the Bahamas, Thi Xoa Nguyen held A♦ Q♣ on a board of K♠ Q♦ 9♠ Q♥ A♠, queens full of aces. Athanasios Polychronopoulos moved all in on the river, and after several minutes she folded her hand face up. He showed pocket aces, aces full of queens; PokerNews and Card Player both reported the hand. Count the cards and the fold looks wrong. Of the 990 holdings her opponent could have had, only 4 beat her and 2 tied. Even among full houses or better she was ahead of 9 holdings to 4. She folded to the betting, not the count. That is the honest exception to the theorem: combinations tell you what is possible, and only the betting tells you what is likely.

When folding a full house is right

Folding needs a read that the other player is rarely bluffing and rarely holds a smaller full house. A clean test is to suppose they hold a full house or better and count who is ahead. A pot-sized bet needs 33.3% equity to call and a half-pot bet 25.0% (the pot odds calculator prices any size). Six worked boards, each counted against all 990 holdings:

Six full houses, every opposing holding counted

Board	Your hand	You hold	Beat you	Tie you	Smaller boats	Equity vs boats
J♠ J♦ 6♥ 6♣ 2♠	J♥ A♠	Jacks full of sixes	1	44	87	82.58%
Q♠ Q♦ 8♥ 5♣ 2♦	Q♣ 8♠	Queens full of eights	0	2	13	93.33%
K♠ Q♦ 9♠ Q♥ A♠	A♦ Q♣	Queens full of aces	4	2	9	66.67%
K♠ K♦ 8♥ 5♣ 2♠	8♣ 8♦	Eights full of kings	15	0	6	28.57%
7♠ 7♦ 7♥ K♣ 2♠	K♦ Q♠	Sevens full of kings	51	84	171	69.61%
9♠ 9♦ 5♥ A♣ K♠	5♠ 5♦	Fives full of nines	21	0	0	0%

"Equity vs boats" supposes the opponent holds only full houses or better and counts a tie as half, so it is your share of the pot, not a win rate. Compare it with the price. Eights full of kings has 28.57%: it beats 975 of the 990 random holdings, and it is still short of a pot-sized bet against that range, though not of a half-pot bet. Fives full of nines has none at all, because every full house on that board is bigger. Few opponents are that strong, so treat this as the limit of the case for folding, not a rule. Every bluff, and every smaller hand they would bet this way, raises your share.

How to use the theorem at the table

The theorem's practical use is in your bets, not your folds. If few players can let a full house go, three things follow.

•When you hold the top full house, bet big. Someone with a smaller one will usually pay you off. Zeebo's post goes further: if you can beat their likely full house, "go allin".
•Do not bluff a player who may hold a full house. However large the bet, the theorem says it will not work. If you cannot beat one, his words are "never ever bluff".
•When you hold a small full house and a tight player raises you, remember the theorem describes you too. People cannot fold because the hand looks too strong, not because the odds always say call.

Other poker theorems: Baluga, Clarkmeister and the fundamental theorem

Zeebo's is one of a family of forum rules of thumb, and the name theorem is equally loose for most of them.

•Baluga theorem: re-evaluate a one-pair hand when you are raised on the turn. ThePokerBank credits it to the Two Plus Two poster BalugaWhale, the screen name of Andrew Seidman.
•Clarkmeister's theorem: heads up and first to act on the river, if the river puts a fourth card of one suit on the board, you should bet. ThePokerBank says it began in limit Hold'em and that it could not find a reliable first source.
•The fundamental theorem of poker: David Sklansky's rule from The Theory of Poker. Whenever you play a hand differently from how you would if you could see your opponents' cards, they gain, and when they play differently from how they would if they could see yours, you gain. It holds for heads-up decisions and not for every multi-way one.