To flop a nut Low draw with this hand, the Flop needs to contain any 2-card combination that has 2 unpaired low cards, exactly one of which may be of the same rank as any one of the hole cards.
This calculation includes the possibility of flopping a made nut Low.
There are 32 cards in the deck below the rank of 9, and 4 of them are in the hole. The number of 2-card combinations that contain 2 cards below the rank of 9 is Comb(28, 2):
This number includes 30 pairs from the ranks 4 through 8, 1 pair of Aces, 3 pairs of Deuces, and 3 pairs of Treys, for a total of 37 pairs that must be subtracted from 378:
From this, subtract the unpaired 2-card combinations both of which match hole cards: 6 A2 combinations, plus 6 A3 combinations, plus 9 2?3 combinations, for a total of 21 combinations that match 2 hole cards:
To include the possibility of flopping a made Low hand, multiply this number (320), by the number of remaining cards in the deck (48), minus the 2 Flop cards that make the
Low hand (2), minus the 6 cards that will pair either of the first 2 Flop cards (48 - 2 - 6 = 40):
There are 12,800 3-card Flops the WILL make a nut Low draw or better:
Total Possibilities = 17,296 - WILLs = 12,800 WILLNOTs = 4,496
Odds of Flopping Nut Low Draw or Better WILLNOTs : WILLs 4,496 : 12,800 Reduce
To express these odds as a probability: WILLs
100 = Probability as %
To flop a nut Low with this hand, the Flop must contain 2 unpaired low cards that do not match any of the hole cards, plus a third low card that may match any of the hole cards, but may not match either of the first two cards flopped.
First you need to calculate the number of 2-card combinations that will make the nut Low draw without counterfeiting any of the hole cards.
There are 20 cards ranked 8 or lower that do not match any of the hole cards. The number of 2-card combinations — Comb(20, 2):
From the ranks 4 through 8, there are 30 possible pairs that must be subtracted from the total to indicate the number of 2-card combinations that will make the draw:
While the third card may counterfeit one of the hole cards and still produce a nut Low, it must not be a card above the rank of 8 and it must not pair one of the other 2 board cards. Thus there are 46 - 20 (number of cards 9 through K) — 6 (number of cards that will pair either of the board cards) = 20 possibilities for the third card. The number of 3-card Flops that WILL make a nut Low to this hand:
With a total of 17,296 possible 3-card Flops, the odds of flopping a nut Low or a Set of Aces:
Total Possibilities = 17,296 - WILLs = 3,200 WILLNOTs = 14,096
Odds of Flopping Nut Low or AAA WILLNOTs : WILLs
14,096 : 3,200 Reduce 14,096 / 3,200 : 3,200 / 3,200 4.4 : 1
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