Note: Not at the old Poker1 site. A version of this entry was first published in Card Player. This entry in the "Aunt Sophie" series covers poker.
Aunt Sophie learns some odds
Aunt Sophie and I continued our discussion in a booth in the coffee shop of the Anaheim Club. Sara had excused herself to go off to her night class in French literature. Since the UC Extension she attended was less than five minutes away, and the class didn’t start for over an hour, I suspected she wasn’t as enthralled with Omaha/8 as was my aunt.
“Tell me,” asked Aunt Sophie, “Omaha drawing odds. The low ones, which always confuse me.”
“Okay,” I assented. “With one card to come, two low cards on the board, and two low ranks in your hand, you will make a low on the river 16/44 of the time. This is true whether or not the board is paired on low cards, and also whether or not your hole low cards are paired or if you have additional cards that pair the board. Let me provide some examples of this draw. The board is K-Q-8-7, and you hold A-T-J-2; you have 16 ways to hit a low: four each of 6-5-4-3. Another: the board is K-Q-8-7, and you hold A-7-2-2; this gives you the same 16 ways. And note that holding a 7 that pairs the board or an extra deuce does not change your draw odds for low here. And: the board is T-9-3-2, and you hold A-K-Q-4. Your 16 winners are any 5 through 8. Now with a board of 3-3-2-2, and A-K-Q-4 in your hand, you still have 16 ways to hit a low. Notice that the board being paired on low cards does not affect your draw odds for low. Last, a board of A-3-9-9, and you hold 7-8-J-K, there are still 16 ways to hit a low, in this case, four each of 6-5-4-2. Sixteen out of 44 are, of course, the draw odds that you’ll hit a low, not that you’ll win with it.”
“Ah bissel more cheesecake you’ll be having?” interrupted Aunt Sophie. “All of my second piece I can’t finish.”
“Aunt Sophie,” I remonstrated, “one piece of blueberry cheesecake, delicious though it is in this fine repository of epicurean delights, is all I can handle.”
“Okay,” she sighed, “just asking,” and swallowed a mouthful of tea to cleanse her palate, and dug in to the delicacy herself.
“Here,” I continued, is a better situation, also with one card to come. This time you have three low ranks (different from the board) in your hand, and there are two low cards on the board, you will make a low on the river 21/44 of the time. Examples of this draw are: the board is K-Q-8-7, and you hold A-T-4-2; or the board is T-9-3-2, and you hold A-K-5-4. As you can see, counterfeit protection, in the form of an extra low card in your hand, is very valuable, in case a low card comes that matches one in your hand. For example, if the board is K-J-7-5, and you hold A-2-9-9, an ace or deuce does not make a low for you, but if you hold A-2-4-9, you still make a low if an ace, deuce, or four comes.
“If one of your low ranks is paired in your hand, you’ll still make a low on the river 20/44 of the time; such as, the board is K-Q-8-7, and you hold A-3-3-2. And if you hold four low ranks not already on the board, you’ll also make a low 20/44 of the time. That’s a confusing one, because at first glance, you might think this would be better than three low cards. Say the board is J-5-5-3, and you hold 8-6-4-2, you have one less chance to make a low than if you are holding A-T-4-2, because you are holding one more low card in your hand (and it’s therefore not in the deck).”
“At least another latte you’ll have,” queried Aunt Sophie, signalling the waitress.
“If you flop the nut low,” I went on, “you will be counterfeited a full 25% of the time! That is, 25% of the time a card will come that matches one of your low cards, unless you are paired in the hole on that card. So, again, counterfeit protection in the form of a third or fourth low card in your hand of a different rank is extremely valuable. This is mostly in the form of additional low cards to preserve your nut or second nut low when you are counterfeited. Paired low cards also help by reducing the number of ways to be counterfeited, and possible making a high hand when you are. In the case of a flopped wheel, being counterfeited still leaves you with a wheel, but the strength of the wheel is greatly diminished.
“If the flop contains two low cards, a low will be possible by the river 75% of the time (assuming no knowledge of any hole cards). If you hold four low cards (not pairing any on the board), a low will be possible on the river 71% of the time. For example, the board is A-A-3, and you hold 4-5-6-7; that gives a 71% chance of one or more of 8-7-6-5-4-2 coming. If you hold no low cards that don’t pair the board, or only low cards that pair the flop, 79% of the time at least one more low card will hit the board to make low possible on the turn or river. Of course, you won’t have a low. You can extrapolate from this that if your hand can go for high only, you will very likely split the pot. Two examples: the board is Q-7-3, you hold K-Q-J-T, there is a 79% chance of one or more of 8-6-5-4-2-A coming; the board is T-8-7, and you hold J-T-9-7, there is a 79% chance of one or more of 6-5-4-3-2-A coming.
“If the flop contains only one low card, a low will be possible 29% of the time. If you hold four low cards, you’ll make this runner-runner low about 25% of the time. Example: with a board of Q-9-3, and you hold A-2-4-6, there is a 25% chance of low making it on the river. If you hold three low cards, you’ll make a low 24% of the time (23% if you are paired in the hole on one of those cards). Examples: with a board of Q-9-3, and you hold A-2-6-T, there is a 24% chance you make a low on the river. And with a board of Q-9-3, and you hold A-2-2-6, there is a 23% chance you make a low on the river. If you hold two low cards, you’ll make a runner-runner low 16% of the time. This could happen with, say, a board of Q-9-3, and you hold A-2-T-T. Notice that in the preceding three examples, a low could be possible on the river, and by pairing both of your cards in the first two and one or both in the last still keep you from having that low. This suggests that backdoor low draws should only be taken if the pot is very large, and even then you generally want protection against counterfeiting. A backdoor low draw can add enough odds to a high drawing hand to proceed after the flop, but is usually not enough on its own to proceed. And that’s a big mistake you’ll see the live ones making, the same folks who play any two suited cards in hold’em and take a card off the deck if one comes on the flop.”
“And do I have to,” she wondered, “remember all these percentages?”
“No,” I stated. “Just follow the precepts extrapolated therefrom.”