Hi Olrik - Hero is stuck at least calling the hand down to the showdown.
That's because if somebody bets and Hero has the winning hand, there will be at least 8 small bets in the pot at the showdown and it will only cost Hero 5 small bets to see the showdown.
Assuming at least one more bet goes into the pot, Hero will thus win at least 8 small bets 21 times out of 45.
+8*21/45=+168/45
Assuming Hero puts at least one bet into the pot each round, in a fixed-limit game, Hero will lose a total of 5 more small bets. And Hero should fully expect to lose 24 times out of 45.
-5*24/45=-120/45
Thus by calling the hand down to the showdown, Hero wins more than he loses. (Hero wins a minimum of +168/45 and only loses a maximum of -120/45).
Therefore Hero has to be in the hand to the showdown. In other words, Hero is stuck in the hand.
Some might argue, "What if somebody bets who would never bluff here?" My reply is simply, "This is poker. Your mother who loves you might bet here to try to steal the pot."
Therefore, steel yourself to be in this pot until the showdown. Steel yourself to fatalistically take the pressure.
You are
not folding this hand. It would be terrible poker to fold this hand. If Jesus Christ Himself bet, confided in you that He held the missing six, told you to trust in Him, and you had never, ever ever seen Him bluff, it would be
terrible poker to fold this hand.
If we're absolutely straight on that point, now the question becomes, "How can you maximize your profit?" And the answer (perhaps as always) is, "It depends on your opponents."
Hero has three players yet to act behind him. If Hero thinks one of them very well might bet if Hero checks, then Hero can check and hope somebody without the six takes the bait and bets. If so, and assuming Hero simply calls, quite possibly Villain will also bet the turn and then also bet the river in an attempt to convince Hero that Villain has the missing six. And in that case, Hero can expect to still lose 21 times out of 45, amounting to a total (in units of small bets) of
-5*24/45=-120/45. But Hero can also expect to win a whopping
+(7+5)*21/45=+252/45.
Or, if Hero thinks nobody will bet, then Hero can still check, and then can still check the turn. And then Hero can bet the river and an opponent with some lower pair might not believe Hero and Hero will win an extra big bet. Hero might very well get raised on the river if he plays the hand this way, and in that case Hero would have to call (remember we're
not folding this hand) and in that case Hero would still lose 21 times out of 45, but for a total of only 4 more small bets, instead of 5. Hero would lose
-4*24/45=-96/45. But Hero can also expect to win
+(7+4)*21/45=+231/45.
If Hero bets on the second betting round and gets called, Hero has to wonder if Villain has the missing six. And then Hero has to check on the third and fourth betting rounds. Villain may bet both of these rounds, or may bet either one of them. Let's say Villain just trickily checks the hand on the third betting round and then bets the fourth betting round after Hero checks again. Remember that Hero may
not fold this hand. (There's one exception. If Hero peeks in Villain's hand and clearly sees for himself the six in Villain's hand, then Hero may fold).
But let's say that Hero bets on the second betting round, Villain who checked ahead of Hero calls, and then Villain checks the third and fourth betting rounds, hoping that Hero bets again. In that case, exactly one more small bet goes into the pot.
Hero would lose
-1*24/45=-24/45. But as opposed to just checking it down all the way to the river, Hero would expect to win
+(+1)*21/45=+21/45.
Thus Hero has favorable odds to call all the way to the river, but does not actually have favorable odds to bet himself!
But don't leap to conclusions.
Although Hero does not have favorable odds to bet himself, on the river it is probably wise for Hero to bet (even with unfavorable odds). And for two reasons. <ul type="square">(1) Nobody holding a six is likely to check on the 2nd, 3rd
and 4th betting rounds and an opponent without the six (but with a pair) may pay off Hero - and if not, Hero doesn't have to show his hand.
(2) No hand is an island. If Hero expects to get paid off on the river when he bets the nuts on the river, then Hero has to bet some non-nut hands on the river. Hero is going to lose 3/45 more often than he wins here, but by betting here, he may collect 45/45 on the next hand he bets with the nuts on the river.[/list]Finally, if you're fairly certain no opponent has a six but a calling station will call you down to the showdown but will not bet himself, it makes sense to bet this hand all the way to the showdown, even with unfavorable odds against the whole field.
I would generally check this hand until the river, and then I would bet the river. I think that's probably a safe bet in my games because not many seated behind me with a six in my games would check both the second and third betting rounds, and even fewer seated ahead of me would check the second, third, and fourth betting rounds. However, someone without a six might call me, especially if I waited until the fourth betting round.
In the previous paragraph I wrote "generally." Sometimes I would bet the second betting round just to see what would happen. And then I'd play from there, depending on what did happen. I could only lose a maximum of
-6*24/45=-144/45. But I would expect to win a minimum of
+(+8)*21/45=+168/45.
The key to playing this hand/flop is staying for the showdown and neither panicking because of fear nor greed.
(Whew, that was longer than I thought it would be). [img]/images/graemlins/tongue.gif[/img]
Buzz