[Buzz]

Mayhap - Would you call if you held the worst possible full house? Yes?

Well, the nut flush is just one step down from the worst possible full house.

Let’s stack the deck to make the final board 6c9c9dTdJc and the hand of the player in the last seat Ac2cAh3h. The player in the last set thus will have the nut flush 10000 times out of 10000 and will only be beaten by an opponent with a full house or better. Let’s give eight opponents random hands. Here’s the data:
Ac2cAh3h....2013 win...................7987 lose....9PP.......nut flush

In this simulation, the player with the nut flush lost 7987 times. In other words,
(1) when you have a nut flush and eight opponents have been dealt cards,
(2) when the board is paired, and
(3) assuming that nobody has folded the cards that would make quads or a full house -
then someone will have a full house (or quads) about four times out of five.

When the final board is 6c9c9dTdJc, in addition to quads, there are six possible full houses. Three of these involve a hand with a pair and three don’t. Following is how each particular type of full house fares when given to the player in the last seat:

10000 trials each. Eight opponents with random hands:
Ac2c6h6d....2903 win...................7097 lose....9PP....underboat
Ac2c6h9h....4466win.....514 tie....5022 lose....9PP....boat
Ac2c9hTh....5967 win....709 tie....3324 lose....9PP....boat
Ac2c9hJh....7160 win....842 tie....1988 lose.....9PP....boat
Ac2cThTd....7990 win...................2010 lose.....9PP....overboat
Ac2cJhJd.....9488 win.....................512 lose.....9PP....overboat
Ac2c9h9d..10000 win.........................0 lose.....9PP....quads

In other words, I ran eight separate nine-handed simulations, 10000 trials each, stacked the deck each time to make the final board 6c9c9dTdJc each time, and also stacked the deck so as to give the player in the last seat a specific hand each time. The specific hands were:
Ac2cAh3h.......(made the nut flush)
Ac2c6h6d.......(made an underboat)
Ac2c6h9h.......(made a boat)
Ac2c9hTh.......(made a boat)
Ac2c9hJh.......(made a boat)
Ac2cThTd.......(made an overboat)
Ac2cJhJd........(made an overboat)
Ac2c9h9d.......(made quads)
Pertinent results are shown above. If I ran the exact same simulations again, I’d expect the numbers to be a bit different, but in the same ball park. If I ran a series of similar simulations (stacked deck with paired board and hand that for sure makes the nut flush or some particular type of full house), I’d expect the numbers to be a bit different, but in the same ball park.

Keep in mind that some of your opponents will fold hands that would have ended up as a full house. And other of your opponents will bet as though they have the nuts when the board pairs.

Bottom line: Do you call a bet on the river with the nut flush when the board is also paired and an opponent bets? Depends, but generally yes, I think you should. I do.

Buzz