Pot odds: a decision, not a result

The other player goes all-in for 10 units on the flop, the first three shared cards, putting every chip they have into the pot. To call, you put in the same 10 units, and the pot will then hold 40 units. You hold four hearts, one short of a flush. Do you call?

The pot after your call holds the other player's bet, everything already in the pot, and your own call.

Leoformer dealer · rules and maths

Players at my tables would argue over a call for ten minutes after the river, the last shared card, had come. The question worth asking was settled before the river: was the price of the call smaller than the share of the pot the cards held? That is a sum you can do at the table, with the pot, the bet and a count of your cards.

The share a call needs

To call is to put in the same amount as the bet in front of you; to fold is to give up the hand and leave whatever you already put in. A call buys a share of one pot, and pot odds are the price of that share. In this lesson's words, the share a call needs is your call divided by the pot as it will stand after your call.

In the example the pot holds 30 units with the other player's bet in it, and the call is 10 units. After the call the pot holds 40 units, and 10 units of that pot is yours.

That makes the share the call needs 1/4, which is 25%.

Odds count the times something does not happen against the times it does. The same reading prices a call: the pot offers 30 units against your 10 units, which is 3 to 1. A call that loses three times for every time it takes the pot comes out even here: three lost calls cost 30 units, and the one pot taken brings in 30 units. So 3 to 1 and 1/4 are the same price written two ways.

Each block is the same amount of money: three blocks are already in the pot, and one block is your call.

Doubling both the pot and the call leaves the share where it was; a smaller pot for the same call raises it. These pots and calls are examples, not a suggested stake. Amounts are in betting units, not pesos: a unit can stand for any amount, and the arithmetic is the same at any scale.

Your share of the pot against the share you need

Your share of the pot is what your hand takes back on average: over every way the remaining cards can come, the part of the pot your hand ends up with, with a split pot counted as your part of it. Lesson 7.7 counts that share from your outs, the cards still in the deck that would put your hand ahead.

Call when your share is larger than the share the call needs, and fold when it is smaller. Against a hand you cannot see, the outs table counts every flush as taking the pot and nothing else as taking it. Read that way, a flush draw with 9 outs holds 35.0% on the flop, with two cards to come (the turn and the river, the fourth and fifth shared cards), and 19.6% on the turn, with one.

• A flush draw on the flop, two cards to come

  Needs 25%Holds 35.0%Call

• A flush draw on the turn, one card to come

  Needs 25%Holds 19.6%Fold

The upright mark on each bar is the share the call needs; the filled part is the share the draw holds. The flop call clears the mark, and the turn call falls short of it.

At an example pot of 30 units and a call of 10 units, the pot after your call holds 40 units. With two cards to come the draw holds 35.0% of it. On average that brings back 13.99 units; less your 10 units call, the flop call adds 3.99 units. With one card to come the draw holds 19.6%. That brings back 7.83 units on average, and less the 10 units the turn call costs 2.17 units. A fold costs nothing more.

Both comparisons assume that no more betting follows your call, which is the case when the other player is all-in. Bets on later streets would change the pot and the price together, and this lesson leaves them out.

The pots here also leave out the room's fee, the rake. At a poker table you play against the other players, the room takes its fee out of the pots, and so the table as a whole gives that fee up.

When you can see the other hand

The outs table treats the other player's two cards as unknown. In practice mode the other hand lies face up, so its two cards are out of the deck as well, and some of the cards that help you help that hand more.

Here the four hearts face three sevens. The two of hearts makes your flush and pairs the board, which gives the sevens a full house, and a board that pairs on the turn or the river fills the sevens up as well. Counted against the sevens card by card, your share falls below the share the call needs.

• The outs table, with the other hand unknown and two cards to come35.0%Call
• Counted against the three sevens, two cards to come24.6%Fold
• Counted against the three sevens after the king of spades on the turn18.2%Fold

The call needs 25% in every row, at the same pot and the same call.

At the example pot of 30 units and call of 10 units, calling against the sevens on the flop costs 0.14 units on average.

Against a hand you can see, the count belongs to that hand alone. Here the count turns the call from the outs table into a fold; against another hand the call could stand, so count each time rather than carry an answer over from the table.

One hand does not grade the decision

A call worth making can lose the pot, and a call not worth making can take it. Here is a call worth making: K♥ Q♥ against A♠ 9♦ on a board of A♥ 7♥ 3♣, with the other player all-in.

Your share of the pot is 38.0%, larger than the 25% the call needs.

At the example pot of 30 units and call of 10 units, the call adds 5.19 units on average.

Below are 100 boards of that same spot, each finished from the cards left, from a fixed seed so that the run is the same every time the page is built.

1. −
2. −
3. −
4. −
5. −
6. −
7. +
8. +
9. +
10. −
11. −
12. −
13. −
14. +
15. −
16. −
17. −
18. +
19. +
20. −
21. −
22. +
23. −
24. −
25. +
26. −
27. +
28. −
29. +
30. −
31. +
32. −
33. +
34. −
35. +
36. −
37. −
38. +
39. +
40. −
41. −
42. +
43. +
44. −
45. +
46. −
47. −
48. −
49. −
50. −
51. +
52. −
53. +
54. +
55. +
56. −
57. −
58. +
59. −
60. −
61. −
62. −
63. +
64. +
65. −
66. +
67. −
68. −
69. −
70. −
71. −
72. +
73. +
74. −
75. −
76. +
77. −
78. −
79. −
80. −
81. +
82. +
83. −
84. +
85. −
86. +
87. +
88. −
89. +
90. +
91. +
92. −
93. −
94. −
95. +
96. −
97. +
98. −
99. −
100. +

The call came out ahead on 40 of the 100 boards, marked with a plus, and behind on the rest, marked with a minus.

Across those boards the call averaged a gain of 6.00 units a call, around the exact average of 5.19 units a call. A run of boards is a sample: its average jumps about from one run to the next, and the exact figure does not move.

What happened in the last hand does not change the next one, and nobody can say how the next hand will go. Strong hands lose pots all the time, even though a stronger hand takes the pot more often.

Mayaformer floor supervisor · manners, money and breaks

Chasing a pot that went the other way, by buying in again or moving to a bigger table to get that money back, is a different decision from a call, and it is never part of this arithmetic.

The money for the session is set before the first hand. When it is gone, the session ends there, and chasing a loss has a lesson of its own.

The lesson on one card

That is the lesson. The share table is on one card: print it, or keep it on your phone.

Practice: call or fold?

Every hand in the practice run stops at the moment you decide. Level 1 is an all-in on the flop with two cards to come, level 2 an all-in on the turn with one, and level 3 an all-in on the flop with a second player already all-in as well, so no more betting can follow. The other hands lie face up, so you count against them.

Nothing is bet here, and the pots and calls are examples. After each answer you see what a call and a fold do on average, and then 100 boards of the same spot.

Check yourself

Mayaformer floor supervisor · manners, money and breaks

Three quick ones. Pick an answer, then check it. No timer, no points: just whether it stuck.

Back to the question we opened with. Call: against a hand you cannot see, four hearts with two cards to come hold more of the pot than the 1/4 the call needs. Against a hand you can see, count against that hand first.

Further reading

18+ under each province’s rules · Play responsibly · the helplines are further up this page

Further reading

• Cracking Aces: Part II · Brian Alspach, Poker Digest (2000)A mathematician’s count of how often a pair of aces holds up against nine opponents.
• Lecture 3: Basic strategy · MIT OpenCourseWare, 15.S50 Poker Theory and Analytics (2015)Outs, the rule of 2 and 4, and pot odds as the share of the pot that a call has to put in.
• Lecture 26: Random walks, the Gambler’s Ruin · MIT 6.042J Mathematics for Computer Science (2000)Why the outcome of one bet does not depend on the bets before it.

How it’s worked out