Many moons ago we played low-limit hold 'em regularly at the Santa Fe in northwest Las Vegas. Every day an older couple would arrive around five p.m. and play until they could obtain adjacent seats, usually the five- and six-seats. They would then leave for an hour while they ate their supper in the restaurant. We don't like partners at the same table, much less side-by-side, but that's not the point of this column. Here we concern ourselves with the effect of players' absences on bad-beat jackpots.
In our local poker room anyone may leave her seat whenever she chooses, but management picks her up after a second "absent" button. The craftier walkers play one hand with a new dealer and extend their allowed absence for a second down, a total of about an hour. This infuriates us not only because it's much harder to hit the bad-beat jackpot short-handed, but if the table hits the bad beat, then an absent player also wins a table-share.
In general, walkers are inferior hold 'em players. They "game" our system in two ways: first, they lose less by staying away from the table for hours and hours; second, they still qualify for a bad-beat table-share. So they wander around the casino, play the penny slots, nosh, have a snort or two at the bar, sit by the pool and people-watch, and/or read a newspaper.
They do that for 59 minutes. Then they seat themselves, take a hand or two with the fresh dealer, and off they go again. They spoil it for the rest of us because their absences materially reduce our table's chance of winning the bad-beat jackpot. Here's proof.
10h, 10d, 4c, 8d, 8h
Suppose the dealer put two pairs on the table like those shown. If a player held pocket tens and another held pocket eights, then having four-of-a-kind beaten by four-of-a-kind would win the jackpot.
We restrict this proof to boards where only quads versus quads are possible. There are 119,232 boards with two pairs and no possibility of aces-full-of-jacks-or-better, nor straight flushes. There are 2,598,960 possible boards, so the probability of a qualifying board is 0.04513.
The four key cards must be among those dealt to the players. For a nine-player table, we obtain the probability from C(4,4)*C(43,14)/C(47,18), which equals 0.01716.
The four key cards must lie in any two player's hands as pocket pairs. That probability is given by 13!!/17!!, or 0.00392.
We multiply the probabilities to conclude that at a nine-player table, the probability of hitting the bad-beat jackpot with quads over quads is 0.00000304, about 329,362-to-1 against. We also did the math for 10 players, 8 players, etc. The table shows the deleterious effect of those (expletive deleted) walkers on short-handed hold 'em tables.
Compared to a ten-player table, it's 20 percent harder to hit the bad-beat jackpot with nine active players at the table. With eight active players it's 38 percent harder to hit the bad-beat jackpot. As if it weren't hard enough already, it's more than twice as hard to hit the jackpot with only seven active players. With only four active players it's seven-and-a-half times (!) less likely than with ten.
While we proved these results for the special case of quads versus quads, this principle is correct whenever both hole cards must play: The fewer the players, the scarcer the jackpots. Therefore we implore, "Walkers of the world, sit down and play!"
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 firstname.lastname@example.org