Worst Bad Beats

[SS] Stan the Stat opened with a question, “What’s the worst possible bad beat you can suffer?”

[UU] Umberto the Unlucky started to respond even before Stan had finished speaking, “Runner-runner two-outer, one-outer. Happened to me just the other day.”

[SS] “But of course, and since we need to know all four hole cards, the odds are two in 45 times one in 44 for a one in 990 chance.”

[SS] “What’s the best hand you can have and lose to this one-tenth of a percent chance?”

[CC] “That’s easy. A Queen-high straight flush losing to the royal flush.”

[LL] “No, your opponent needs one of the those cards, so it’s a Jack-high straight flush”, Leroy the Lion adjusted. “You could have 8♥7♥ against the A♥ and any other card on a J♥10♥9♥ flop. Your opponent needs both the K♥ and Q♥ to hit.

[SS] “Right. What’s the worst hand you can lose to?”

[LL] “Well, there are only two types of hands your opponent can be drawing to: either three-fifths of a straight flush or two-fourths of quads. So the answer has to be four Threes. If you have pocket Twos against pocket Threes and flop quads vs. two pairs with no straight flush possible, your opponent needs runner-runner Three-Three to beat you.”

[SS] “Close but no cigar.”

[CC] “So it must be four Twos, not four Threes. Your hand could just be a full house, not quads.”

[SS] “Exactly. For example, if you have Ace-King against pocket Twos and the flop is the Ace-Ace-King, then your opponent needs running Twos to beat you.”1

[SS] “So what’s the worst hand you can have?”

[LL] “Well, that previous case works all the way down to Threes full of Fours, but wait… If the board is unpaired, your opponent may need a straight flush to win against a flush. So some low flush.”

[SS] “Very good, but how low?”

[LL] “Your opponent has to have only one way to make the straight flush, so put the 4♠3♠2♠ on the table and give him the 7♠ and any non-spade that doesn’t give him a pair. Then I can have the 9♠8♠ for a flopped flush, and he’ll need the 6♠ and 5♠ to win.”

[SS] “Well done!”

[EE] Elias the Eagle had wandered over during the discussion and finally chimed in, “But what if you aren’t heads up?”

[SS] “Ooh…”

[EE] “I can have a hand as bad as three Twos and have nine opponents each of whom could beat me with a runner-runner, two-outer, one-outer, and for a three-way chop no less!”

[LL] “How?”

[EE] “If I flop a set of Twos on an Ace-Six-Two rainbow flop, I could have nine opponents using up three of the four Fives, Fours, Threes, and three other denominations. For example, K5, K5, K5, Q4, Q4, Q4, J3, J3, and J3. The Fives need the remaining Four and Three, the Fours need the last Five and Three, and the Threes need the case Five and Four for a Six-high straight.”

[SS] “Or even better, a mere pair of Sevens. 74 vs. 32 vs. 32 vs. 32 on an Q87 rainbow flop can only lose to the last Three and the last Two.”

[LL] “You can get all the way down to a pair of Fours! 54 against four players with A6 and three players with 32 and a K94 with no flush draw does the same trick.”

Footnotes:

  1. Note that if you have a full house holding a pocket pair, quads can beat you, as discussed in The Nuts.
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