Simple sequences of betting in poker

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HoldemTo illustrate the mechanism of the betting and the corresponding mathematical reasoning, here it is assumed that:

  • eight players play poker with 52 cards. The initial pot is an amount P;
  • Alice opens, Bob calls, and the others fold;
  • the openings are only “pot” or “half-pot.”

In the example shown, Alice has a pair and exchange two cards, while Bob has a drawing and will exchange one card only. This hypothesis has the advantage of greatly simplifying Alice and Bob options: Only Alice can have two pair or three of a kind, and only Bob can have a straight or a flush. In addition, to 52 cards, an unimproved pair coincides with the opening half-pot, which greatly simplifies the interpretation of the opening of Alice.

Pair against drawing

One of the most basic scenarios are: Alice opens the pot, Bob calls. Alice exchange two cards (pair), Bob one. In absolute terms, having exchanged two cards, Alice can have:

2 cards Nothing Pair Two pairs 3 of a kind Straight Flush Full 4 of a kind Royal flush
proba 0,0 % 67,7 % 15,1 % 15,4 % 0,0 % 0,0 % 1,3 % 0,5 % 0,0 %
cumul 100,0 % 32,3 % 17,2 % 1,8 % 1,8 % 1,8 % 0,5 % 0,0 % 0,0 %

For 52 cards, Alice will actually have her initial pair (75%), two pair (17%) or 3 of a kind (7.8%). If Bob drew one card, it can have in the absolute:

1 card Nothing Pair Two pairs 3 of a kind Straight Flush Full 4 of a kind Royal flush
proba 32,9 % 15,0 % 37,4 % 0,0 % 3,9 % 7,0 % 3,5 % 0,2 % 0,0 %
cumul 67,1 % 52,0 % 14,6 % 14,6 % 10,8 % 3,7 % 0,3 % 0,0 % 0,0 %

If Alice has not improved her pair (three out of four), his chances of winning are greater than 33% (as Bob has nothing in 32.9% of cases), and less than 48% (as Bob has more than a pair in 52% of cases). Alice can not reasonably open only half a pot. Bob then knows that Alice has a simple pair, and has not improved. If he has nothing or a small pair, he passes. If he himself has a sufficient pair, he can call to see (in 3-4% of cases), if he has more than a pair it is enough to call to win.

So the significance of this exchange is implicitly:

  • (Alice) Opening the pot (I have at least a high pair).
  • (Bob) Call (at least I have that).
  • (Alice) Two cards (it is a pair or three of a kind).
  • (Bob) A card (it’s a draw or two pair).
  • (Alice) Opening at half-pot (my pair did not improve).
  • (Bob) Call (similar pair, or winning hand not more) or fold (not presentable pair).

Improved pair against drawing

If Alice has improved its pair (one in four), it can open the pot (see previous discussion). In this case, Bob can exclude a pair in Alice‘s hand, and knows that she has in front of him, in the remaining cases, the following distribution:

!2 cards Nothing Pair Two pairs 3 of a kind Straight Flush Full 4 of a kind Royal flush
proba 0,0 % 0,0 % 46,7 % 47,7 % 0,0 % 0,0 % 4,1 % 1,5 % 0,0 %
cumul 100,0 % 100,0 % 53,3 % 5,6 % 5,6 % 5,6 % 1,5 % 0,0 % 0,0 %

The optional double pair is necessarily high, as the pair has justified opening the pot. To call, he must be able to win on its gain against a two (33%). If he has himself only nothing or a pair, he folds. If he has two pair, he also folds, unless its strongest pair is itself a level that warrants the opening (in ¼ of cases). Otherwise, it can call having his normal chance to win.

The characteristic exchange then implicitly means:

  • (Alice) Opening the pot (I have at least a high pair).
    (Bob) Call (at least I have that).
    (Alice) Two cards (it is a pair or three of a kind).
    (Bob) A card (it’s a draw or two pair).
    (Alice) Opening the pot (my pair has improved).
    (Bob) Call (double pair comparable or winning hand not more) or fold (less than a strong two pair).

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