# Simple sequences of betting in poker

posted in: Poker strategy |

To 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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Simple sequences of betting in poker
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