There have been a couple interesting threads on rec.gambling.poker about what the theoretical worst bad beat in poker is.

For those of you new to poker, a "bad beat" is when a hand that is very poor and has little chance of winning gets its miracle card(s) to win the pot from a better hand. An example from holdem would be where a player is holding A♠A♣ is up against a player with 2♠3♣. The aces are a huge favorite. Then, say the flop comes A♦7♣2♦. Now the aces are even more of a favorite. But by some miracle the turn and the river bring 2♣2♥, giving the underdog four of a kind, thus beating the aces. That would be a very bad beat.

I quantify a bad beat as a situation in which a player with very low Expected Value gets his cards to win (or occasionally tie) the pot. I'm going to measure the E.V., or average E.V. for multi-round situations, to try to come up with the worst bad beats possible for every game of poker. This list will be incomplete, with games coming as I get to them.

To get the numbers shown below, I have mostly used the pokenum program, which you can use online at this link.

Why does this matter? Well, it doesn't really. It's mostly an academic exercise. The next time someone says "I got hit with the worst bad beat EVER!" - something commonly heard in cardrooms everywhere - I'd like to be able to say "really?? You had a made king-high straight flush in five card draw and got outdrawn by a guy who redrew five cards to the only remaining live royal flush in the deck??"

Regarding the use of dead cards, some people on the R.G.P thread didn't use dead cards in coming up with bad beats, I guess because they thought of them as "cheating". I use dead cards here, as long as they could feasibly be known to be dead by the players during the course of a normal (non-misdealt or misplayed) game. So for hidden games like 5-card draw, hold'em and omaha, I don't use dead cards; for stud games and the like, I use them provided the quantity doesn't exceed what players could see in a normal, full-table game.

Also, I'm not including "chop only" beats where the only outs the underdog has are to a split pot. If you split the pot with the underdog, it wasn't a bad beat :)

The worst beat in hold'em after the flop is where a player has exactly two cards ("perfect-perfect runner-runner") to win the pot. There are many possible situations like this in hold'em -- runner runner straight flush, runner runner quads being the most common.

If we combine this situation with the worst pre-flop heads-up situation, we get the worst possible bad beat possible in hold'em.

Hand	Pre-flop EV (Wins)	Flop	Flop EV (Wins)	Turn	Turn EV (Wins)
K♠K♣	0.949 (1,612,287/1,686,112)	K♦7♣2♥	0.999 (989/990)	2♦	0.977 (43/44)
K♥2♣	0.051 (73,825/1,686,112)		0.001 (1/990)		0.023 (1/44)

Similar to hold'em, the worst beat in omaha after the flop is where a player has exactly two cards that must come to win the pot.

If we combine this situation with the worst pre-flop heads-up situation where the underdog still has outs to win, we get the worst possible bad beat possible in omaha. I'm not yet 100% sure this is the worst possible beat in omaha, but it's the best I've come up with so far.

Hand	Pre-flop EV (Wins)	Flop	Flop EV (Wins)	Turn	Turn EV (Wins)
A♠A♥K♠3♥	0.938 (1,017,155/1,086,008)	A♦A♣5♠	0.999 (819/820)	6♠	0.975 (39/40)
3♦3♠3♣2♠	0.062 (66,463/1,086,008)		0.001 (1/820)		0.025 (1/40)

The worst possible bad beat on the end in 7-card stud is a situation in which the only way the underdog can win by getting one perfect card, while the favorite also catches the only card in the deck that does not improve their hand. Here's one such situation (from Perry Friedman) where the underdog has to catch the only remaining ace while the favorite catches the five of spades:

Hand	Dead Cards	EV (Wins)
Q♣Q♦J♣T♣9♣4♦	A♠A♥7♦7♥6♦6♥5♦5♥3♥3♠2♥2♠	0.999 (755/756)
A♦8♠7♠6♠3♦2♦		0.001 (1/756)

The worst possible beat in seven-stud hi/lo is similar to that of seven-card stud hi-only in that it's also 755:1.

Hand	Dead Cards	EV (Wins)
K♦K♠8♣7♣6♣5♦	A♠A♥7♦7♥6♦6♥5♦5♥3♥3♠2♥2♠	0.999 (755/756)
A♦8♠7♠6♠3♦2♦		0.001 (1/756)

Similarly to seven-card stud, a five-card stud bad beat occurs where one player catches perfect while the other catches their loser. Here's an example (from Tom Weideman) where the underdog has to catch the case king while the favorite has to catch the only remaining deuce:

Hand	Dead Cards	EV (Wins)
K♦K♥4♥3♠	2♣2♦	0.9994 (1721/1722)
K♣5♦4♠2♥		0.0006 (1/1722)

OK, so it's not REALLY poker, but the worst beat possible in Pai Gow is nonetheless interesting and here for completeness. If you are dealt a royal flush with a pair of kings, using the bug (joker) to complete your royal, it is possible for you to still lose! The dealer has one way to beat you: with a royal flush of the suit of the remaining king, plus the case aces. There is only one way to make this hand.

A♠K♠Q♠J♠? K♥K♦

vs

A♣K♣Q♣J♣T♣ A♥A♦

Now that's a bad beat and a half. Assuming that you are playing the common version of Pai Gow where the dealer shows one card, and you see the dealer has one of the required cards to beat you, the odds of the other 6 cards being exactly what the dealer needs are 1 in 7,059,052.

If you are playing a version of Pai Gow where you don't see one dealer card, the odds of the dealer beating you are much worse -- an astronomical 1 in 45,379,620.

If you play Pai Gow for long enough, you will eventually get beaten this way. But you would have to play for an extremely long time.

In compiling this list, I've come across some interesting scenarios. These are questions to exercise your knowledge. You should be able to independently verify your answers, but if you have questions you can e-mail me (address on the left).

In Omaha hi-only, A♠A♥2♥2♠ is a huge favorite (EV of 0.879) over 3♠3♦3♥2♣. Out of the 1,086,008 possible boards, the two hands tie on just 900 (0.08%) of them. What boards do they tie on?