Outs, Odds, Probability, Equity and EV

[Drydoc]

Hello all,

I began learning Texas Hold’em a year ago, but had to take a detour. I’m back in the study of the game, however, starting over and relearning…everything. In effort to understand the concepts accurately, I’m posting this for clarification. I realize I’m lumping a lot into one thread (and you don't like it), however, by keeping this in one place it will be much easier to relate each concept to the other. My apologies for dragging you through the basics…yet again.

Outs, Odds and Probability

Beginning with and expanding Matthew’s “Outs” table on page 23 of ITH we have the following;

-----------------------------------Number of bets in the pot
----------------------------------1nB------2nB------3nB-----4nB
Outs---Odds-----Prob---Bet/Call--Raise--Reraise--Cap
1------46 to 1----2.13%-----47--------93-------139------184
2------23 to 1----4.17%-----24--------47--------70--------92
3------15 to 1----6.25%-----16--------31--------46--------61
4------11 to 1----8.33%-----12--------23--------34--------45
5--------8 to 1---11.11%------9--------17--------25--------33
6--------7 to 1---12.50%------8--------15--------22--------29
7--------6 to 1---14.29%------7--------13--------19--------25
8--------5 to 1---16.67%------6--------11--------16--------21
9--------4 to 1---20.00%------5---------9---------13--------17
10----3.7 to 1---21.28%------4---------8---------12--------15
11----3.3 to 1---23.26%------4---------7---------10--------14
12------3 to 1----25.00%------4---------7---------10--------13
13----2.6 to 1----27.78%------3---------6----------8--------11
14----2.4 to 1----29.41%------3---------5----------8--------10
15----2.1 to 1----32.26%------3---------5----------7---------9
16----1.9 to 1----34.48%------2---------4----------6---------8
17----1.8 to 1----35.71%------2---------4----------6---------8
18----1.6 to 1----38.46%------2---------4----------5---------7
19----1.5 to 1----40.00%------2---------4----------5---------7
20----1.3 to 1----43.48%------2---------3----------4---------6

To clarify the additional columns, take this example…

Example 1:

I’ve determined I have 5 outs (considering discounts and counterfeits) to draw to the winning hand, not just improve. Therefore, I will improve to the winning hand 1 time and NOT improve to the winning hand 8 times, in 9 hands (similar to this), and therefore have 11.11% chance to win the hand.

1) To make the decision to draw there needs to be at least 9 nBets (including mine) in the pot to make the decision to draw correct.
2) If I expect to put 2 nBets in the pot, I now need 17 nBets (including mine) in the pot for drawing to be correct.
3) If I expect to put 3 nBets in the pot, I now need 25 nBets (including mine) in the pot for drawing to be correct.
4) Same conclusion for a capped pot, 4 nBets requires 33 in the pot to be correct.
5)Regardless of how much of the pot I contributed in earlier rounds, I only count the number of bets required of me in the current round when determining pot odds.

Clarification/confirmation please:

First, these odds are “to improve to the winning hand” and not simply to improve the hand. Correct?

Second, taking the same example above with 5 outs. If there are already 8 nBets in the pot at the time I must make a decision, my bet (making 9 nBets in the pot) is correct IF I will not have to put further bets into the pot (in the current round). Correct?

Third, using same example. If there are already 7 nBets in the pot at the time I must make a decision, but now I expect one or more players behind me to contribute a bet, my bet is correct if I will not have to put further bets into the pot (in the current round). (An example of “implied odds”, correct?)

Finally, if there are already 9 nBets in the pot but I expect a raise behind me, there now needs to be 17 nBets in the pot (by the end of the round) for my contribution of 2 nBets to be correct. Accurate?

How am I doing so far?

Equity, EV, and Value betting.

I understand equity to be the % on average we will win the hand at showdown. Simple, but not the whole picture. Let's look at the next example.

Example 2: (figures calculated with Pokerstove)

Player1 holds KcKd = 60.335% or .60
Player2 holds KhQs = 7.12% or .07
Player3 holds AsTs = 32.54% or .32

Player1’s equity is .60 which means that for every $1.00 he puts into the pot he can expect to win $.60 of his bet, $.60 of Player2’s bet, and $.60 of Player3’s bet, or $1.80 total…a 1-1.8 ratio, positive equity.

Player2’s equity, following the same reasoning is a 1-.21 ratio, negative equity.

Player3’s equity is 1-.96, slightly negative.

Naturally, in real play we do not know the other players holdings which eludes to discussions on types of players, playing ranges and such to allow us to at least narrow down what he has.

In application we use this ratio to determine what action is appropriate during a given round. If our equity is positive as it is with Player1 above, our bets have “value” and it’s to our advantage to get as much money in the pot as we can. In this example we should be betting/raising, correct?

Player2/3 should be checking, or avoiding putting more money in the pot because their equity is negative. This isn’t to suggest they fold to a bet/raise. If they are faced with a bet or raise, they must fall back on the Odds table, calculating the chance to draw to the best hand vs. the pot size to current bet.

Summary/Conclusion

Odds/Probability tells us when we should be trying to draw to the winning hand. If the ratio of Pot-To-Bet (Pot Odds) is greater than the Odds to improve to the best hand, it is correct to draw.

Equity and EV tells us when we should be checking, betting or raising. When our equity is negative we should be checking and possibly folding. When we expect our hand has positive equity or value, we should bet or raise the hand.

These two concepts, while mathematically related, represent independent concepts, each of which needs consideration.

Ok, this post is a bit long, but I’ve tried to trim out the fat leaving enough for clarity. This is my understanding of the concepts. Please point out any inaccuracies and/or elaborate on anything that needs it.

Thanks in advance for any thoughts here.

[Primitive]

Wow, long post :

Looks like you got the concepts down just fine.

Just to clarify:

[Drydoc]

[riverme_now]

So far so good.