Our local poker room pays a hold 'em bad-beat jackpot whenever aces-full-of-jacks or better loses to a higher hand, providing both players' hole cards play. During every hand the players always look at the tableau for a possible bad-beat jackpot. Yesterday, our dealer laid out the tableau shown which stimulated vigorous controversy.
3h-4h-5h-6h-8c
Our dealer opined that a bad beat would occur if someone held Ah 2h and someone else held 7h 8h. She erred. You see why, don't you?
Let's get the math out of the way. There are 7 four-card sequences of this type, times 4 suits, times 44 fifth cards, a total of 1,232 boards out of C(52,5) possible, a probability of 0.000474, or odds of 2,108.5-to-1 against. The four cards at the top and bottom have to be among the 18 dealt to all nine players, a probability given by C(43,14)/C(47,18), or 57.3-to-1 against. The odds against the four cards being in two players' hands as needed are 254-to-1. The odds against a "bad beat" of this type are 31 million-to-1.
When I told our dealer that it wouldn't be a bad beat at all, she averred that of course it would since Ah 2h 3h 4h 5h would be a straight flush and 4h 5h 6h 7h 8h would also be a straight flush. A straight flush beaten by a straight flush would be a bad beat if there were more than $30 in the pot.
"Nope," I said, "it doesn't matter how big the pot would be because the ace wouldn't play." That created quite a stir. Many of the eight other players agreed with her, and strongly.
She dealt the next hand, and yet the conversation again returned to that "bad-beat" hand. They asked me why it wouldn't be a bad beat. "Because a six-high straight flush is higher than a five-high straight flush," I responded. Most remained unconvinced. I didn't have the wit to explain it properly, so it took some time before the light dawned on the sharper knives in the drawer.
Here's what I should have said. "This isn't Omaha where you must play two from your hand; this is hold 'em where you play none, either, or both of your hole cards to make the best hand you can.
"Because you can play none, one, or two hole cards, each hold 'em player has 21 possible hands. You don't get to choose which hand you play; you must play the best hand from the 21 possible.
"In the case shown, playing the ace-deuce would obtain the player a five-high straight flush. Playing just the deuce would obtain a six-high straight flush. He must play his best hand, i.e., the six-high straight flush, so his ace doesn't play." QED
This proves that there is no suited, zero-gap, four-card sequence that would allow a bad beat when both hole cards must play. By extension, there is also no suited, zero-gap, five-card sequence that would allow a bad beat under the rule that both hole cards must play.
One- and two-gap four-card sequences do exist which allow straight flush vs. straight flush bad beats. The different types are:
3h 4h 5h 7h 8c 10c Jc Ac
9s 10s Js As 8d 10d Jd Kd
5h 8h 9h Jh 5s 8s 9s Qs
Each type has seven possible sequences so the odds against each type are 31 million-to-1, per the math above. The probability of a bad beat of this type, a straight flush beaten by a higher straight flush with four of the cards on the tableau, is 0.0000001914, or odds against of 5,225,958-to-1.
Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold 'Em, on sale at amazon.com & kokopellipress.com. E-mail your Hold 'Em questions to richardburke@comcast.net
