LO8 - When is it correct to chase runner-runner for the nut low

cornhuskerfan · #1 08-02-2006, 12:08 PM

A situation came up several times in Vegas that got me thinking.

The game was $4/$8 Omaha8 with numerous bad players that played too many hands and chased too many long shots. I can literally remember playing against two different players who did not fold a single hand, not one, preflop in separate 3 hour sessions. So I got to thinking…., how big does the pot need to be on the flop to consider chasing runner-runner low cards in hopes of getting ½ (or even ¼) of the pot? Please check my logic and let me know where I screwed this up.

The Situation:
You hold A 2 9 K rainbow. The board reads J [img]/images/graemlins/heart.gif[/img] J [img]/images/graemlins/club.gif[/img] 8 [img]/images/graemlins/heart.gif[/img]. You assign a 0% chance of holding any high equity.

The Math:
There are 25 cards that could come on the turn that are helpful (4 each of 7s, 6s, 5s, 4s, 3s). The probability of catching a good card on the turn is 20 cards out of 45 unseen cards - 0.444. Then, you’ll need to catch 16 good cards on the river out of 44 unseen. 16/44 = 0.364. So the probability of catching good on both the turn and the river is 0.444 x 0.364 or 0.162. [img]/images/graemlins/confused.gif[/img]

Split Pot:
If I catch good on both the turn and the river, I’ll get a portion of the pot – either ½ the pot or ¼ of the pot (or maybe 1/6 of the pot – yuk). Since chasing runner-runner low to get a portion of the pot is a long shot (0.162), the pot will need to be pretty big. To create a big pot, it is necessary to have many contestants. With a large number of contestants in an Omaha8 game, the chance of another player holding the nut low as well must be considered. I used Alspach’s Poker math to estimate this probability at 0.3 (<http://www.math.sfu.ca/~alspach/mag8/>)
Therefore there is a 0.7 chance of winning ½ the pot and a 0.3 chance of wining a ¼ of the pot. So, if I catch runner runner low, I expect to get ((0.7 x ½) + (0.3 x ¼)) or 0.425 of the pot.

The Flop::
There is a 0.162 chance of catching runner runner by the river for me to win a 0.425 portion of the pot
0.162 x 0.425 means I have pot equity of about 7%. Therefore, I’d need the pot to be “huge” to make a call chasing runner runner low. I’d need 15 small bets to be in the pot to justify a 1 bet call. To get this number I took the number of bets (1) divided by the probability of winning (0.162) x the proportion of the pot that you’d win (0.425).
To get 15 bets in a pot preflop, I’d need 5 people to put in 3 bets or 8 people to put in 2 bets or 3 people to put in 5 bets (which is a cap in Vegas limit poker).
It gets much worse if you are facing a raise (or worse yet, 3 bets cold on the turn)
If you are facing 2 bets on the flop and still have 7% pot equity, you’d need there to be 29 bets (6 people in a capped preflop pot). <<2 bets divided by the probability of winning (0.162) * the proportion of the pot that you’d win (0.425)>>
If you are facing 3 bets cold, you’d need there to be around 45 bets (9 people in a capped preflop pot). <<3 bets divided by the probability of winning (0.162) * the proportion of the pot that you’d win (0.425)>>
The Bottom Line: you can call on the flop chasing runner runner nut low, if the pot is huge (15 or more small bets) and the flop betting is not raised and you don’t expect it to get raised (i.e., you are in last position).

The Turn:
Now, I have the same problem on the turn when I catch good. Do I continue to chase a low card to the river to get ½ or ¼ of the pot? The math is about the same. There are now 16 good cards out of 44 or 0.364. So, 0.364 – probability of making a hand x .425 – portion of the means I have about 16% pot equity So, we’d need 7 big bets in the pot to call 1 big bet on the turn <<1 bet divided by the probability of winning (0.364) * the proportion of the pot that you’d win (0.425)>>
From the flop discussion above, we must have at least 15 small bets preflop to call a flop bet. You can only call on the flop if it is checked or there is 1 bet; therefore, there are a least 7 big bets on the turn, so the 1 bet call on the turn is automatic if you catch good.
But what if you are facing a bet and a raise on the turn? With 16% equity, you’d need about 13 big bets to justify calling 2 bets on the turn. <<2 bets divided by the probability of winning (0.364) * the proportion of the pot that you’d win (0.425)>>
If you are facing 3 bets cold on the turn, you’d need 19 big bets to justify a call with 15% equity. <<3 bets divided by the probability of winning (0.364) * the proportion of the pot that you’d win (0.425)>> This would be a gigantic pot ($150 pot in a $4/$8 game).
The bottom line: You can call 1 bet on the turn if the pot is unraised. You can call 2 bets on the turn if there are 19 bets in the pot. To get 19 big bets in a pot and still have the correct calling odds on the flop and the turn, you’d need about 8 players in the hand each having put in 2 small bets preflop, 1 small flop bet, and 1 big turn bet…..not likely. You can’t really play for a raise on turn.

The River:
The river plays itself. If you catch good, call (bets and raises). If not, fold. You have about .425% pot equity <<1 bet divided by the probability of winning (1.0) * the proportion of the pot that you’d win (0.425)>> The pot has at least 7 big bets (the criteria for calling on the turn), so you are paying 1 bet to get 42.5% of at least 7 big bets.
What if 2 jackasses cap it on the river? You have 42.5% equity in a pot that has at least 7 big bets. If 2 jackasses cap it, there will be 21 big bets in the pot when you are faced with your last call (5 bets from jackass #1, 5 bets from jackass #2, 4 called bets from you, plus 7 bets to start with). Since you “own” 42.5% of this pot (21 bets x 42.5% = 9 bets that you own) and you only have to call 1 last bet, the call is automatic.

The bottom line: Call all bets and raises on the river when you hold the nut low. Duh! [img]/images/graemlins/tongue.gif[/img]