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Chris Fargas on Twenty-One Outs Twice wrote an interesting article on a Razz hand that he, quite bravely, recounted despite him ballsing it up. Well, he didn't balls up the analysis, but he did get the numbers wrong.
To recap, after four cards, one guy had four to a seven (although Chris could not know this for sure), Chris had four to a 98, and the two other players had caught paint on fourth street. Chris had been bumping it under the not unreasonable theory that he was getting three-to-one for his money and he reckoned that he was better than three-to-one to win it.
In fact, he had the worst of it. He was the dog of four.
It's interesting to break this down a bit.
Suppose the cards are face up and you have 9876 versus A23J after four cards (A2345 is the best low). Which hand would you prefer?
Most players would, I think, correctly plump for the A23J.
Now, suppose you had 982A vs K32A. Which do you prefer now?
Suddenly, the balance of power shifts, and people start preferring the four to the 98.
But this is a relatively simple hand to break down mathematically. Rather than make it the best five-card low hand in a seven-card game, we can despatch the Aces and twos from both hands and make it the best three-card low hand in a five card game.
So, we have 98 vs K3, best three-card low wins with three cards to come. Aces and twos not allowed
I don't have a poker calculator to hand, but clearly if hand B gets two cards seven or lower, hand A needs three cards seven or lower. If hand B gets two cards 8 or lower, hand A needs two 'good' low cards to snap off the 3. If hand B gets two cards 9 or lower, hand A is also likely to need two good low cards to avoid being beaten by the three.
Looking at that I would instinctively make the K3 something like 53% to 55% favourite.
Now, there's one drawback to this analysis, in that betting does not stop on fourth street. It's not enough just to know that the K3 is favourite. We also need to know how things are likely to shift on card five.
If both cards are dogs (ten and above, pairing the hands, etc), 98 becomes favourite, about 60/40?
If both cards are good (seven or below, not pairing the hands), K3 becomes favourite, about 65/35?
If 98 hits and K3 misses, 98 become somthing like 75/25 favourite
If K3 hits and 98 misses, K3 becomes something like 70/30 favourite.
All these are rough estimates in my head, I hasten to add.
So, in none of the above cases is either player going to fold. Card five is likely to result in bet, call, from one side or the other.
If we put the above scenarios as roughly equal in likelihood (25% each) and combined them with similar ofdds for card six (in fact, the hands are more likely not to be helped than helped on card six, but this doesn't change the underlying maths to a significant degree), then we have 16 possible combinations for cards five and six, but six of them (1/1, 1/2, 2/2, 2/1, 3/4, 4/3) restore the status quo.
A further eight (1/3, 1/4, 2/3, 2/4, 3/1, 3/2, 4/1, 4/2) put us into bet, call mode, while just two (3/3, 4/4) put us into bet, fold mode.
The debate here could be that 3/1 puts us into a bet/fold situation where the K3 should quit, while 4/1 keeps us in a bet/call situation where 98 does not quit.
But, on the whole, the fact that there is further betting doesn't really change the situation (apart from increasing the size of the pot). People seem to think that, because the K32A needs two cards for a 'made', it is at a disadvantage to the 982A which needs just one card for a 'made' hand. And, if the game were No Limit or even pot limit, an argument could be made for this.
But in limit, there is no way that there can be insufficient money in the pot to make it wrong for the K3 to call.
it is this that gives us the Sklansky line that the 98 should check here to keep the pot small, thus making it wrong for the King to call on fifth street if he misses. However, the point is that the "leading" hand isn't really leading at all. The King should be betting, not the nine.
To recap, after four cards, one guy had four to a seven (although Chris could not know this for sure), Chris had four to a 98, and the two other players had caught paint on fourth street. Chris had been bumping it under the not unreasonable theory that he was getting three-to-one for his money and he reckoned that he was better than three-to-one to win it.
In fact, he had the worst of it. He was the dog of four.
It's interesting to break this down a bit.
Suppose the cards are face up and you have 9876 versus A23J after four cards (A2345 is the best low). Which hand would you prefer?
Most players would, I think, correctly plump for the A23J.
Now, suppose you had 982A vs K32A. Which do you prefer now?
Suddenly, the balance of power shifts, and people start preferring the four to the 98.
But this is a relatively simple hand to break down mathematically. Rather than make it the best five-card low hand in a seven-card game, we can despatch the Aces and twos from both hands and make it the best three-card low hand in a five card game.
So, we have 98 vs K3, best three-card low wins with three cards to come. Aces and twos not allowed
I don't have a poker calculator to hand, but clearly if hand B gets two cards seven or lower, hand A needs three cards seven or lower. If hand B gets two cards 8 or lower, hand A needs two 'good' low cards to snap off the 3. If hand B gets two cards 9 or lower, hand A is also likely to need two good low cards to avoid being beaten by the three.
Looking at that I would instinctively make the K3 something like 53% to 55% favourite.
Now, there's one drawback to this analysis, in that betting does not stop on fourth street. It's not enough just to know that the K3 is favourite. We also need to know how things are likely to shift on card five.
If both cards are dogs (ten and above, pairing the hands, etc), 98 becomes favourite, about 60/40?
If both cards are good (seven or below, not pairing the hands), K3 becomes favourite, about 65/35?
If 98 hits and K3 misses, 98 become somthing like 75/25 favourite
If K3 hits and 98 misses, K3 becomes something like 70/30 favourite.
All these are rough estimates in my head, I hasten to add.
So, in none of the above cases is either player going to fold. Card five is likely to result in bet, call, from one side or the other.
If we put the above scenarios as roughly equal in likelihood (25% each) and combined them with similar ofdds for card six (in fact, the hands are more likely not to be helped than helped on card six, but this doesn't change the underlying maths to a significant degree), then we have 16 possible combinations for cards five and six, but six of them (1/1, 1/2, 2/2, 2/1, 3/4, 4/3) restore the status quo.
A further eight (1/3, 1/4, 2/3, 2/4, 3/1, 3/2, 4/1, 4/2) put us into bet, call mode, while just two (3/3, 4/4) put us into bet, fold mode.
The debate here could be that 3/1 puts us into a bet/fold situation where the K3 should quit, while 4/1 keeps us in a bet/call situation where 98 does not quit.
But, on the whole, the fact that there is further betting doesn't really change the situation (apart from increasing the size of the pot). People seem to think that, because the K32A needs two cards for a 'made', it is at a disadvantage to the 982A which needs just one card for a 'made' hand. And, if the game were No Limit or even pot limit, an argument could be made for this.
But in limit, there is no way that there can be insufficient money in the pot to make it wrong for the K3 to call.
it is this that gives us the Sklansky line that the 98 should check here to keep the pot small, thus making it wrong for the King to call on fifth street if he misses. However, the point is that the "leading" hand isn't really leading at all. The King should be betting, not the nine.
