Micro Limit Math Thread

[Delphin]

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But what I'm looking for is a set of forumlae to figure this out, instead of trying to run through the 14 possible combinations myself. I haven't seen any set of formulae or a table that takes strictures like, but not limited to, my original post and helps count the possible combos on-the-fly.

Of course, being able to do stuff like this is useful for determining the true number of outs you have.

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The way you determine this is by counting. There is a whole branch of mathematics dedicated to counting called combinatorics. Grab a book on it if you want to learn more about it and apply it to poker. I'm not sure that you will find what you are looking for. There aren't any magical formulas that will tell you how many outs you have in every situation. You have to just count them.

In the situation you mentioned, you want to know how many ways your opponent can have the ace of hearts and pair the board. There are four cards on the board that have 3 unseen cards that make a pair, the fifth card on the board pairs a card in your hand, so there are only two cards out there. The number of ways he can pair the board is:

4*3+2 = 14

Let say you were interested in how this compares to the number of hands that pair the board with any [img]/images/graemlins/heart.gif[/img]. There are 8 hearts he could hold including the A [img]/images/graemlins/heart.gif[/img] and he could hold any of the aforementioned cards that pair the board:

8*(4*3+2) = 8*14 = 112

So some multiplication can save you from just listing out all the possibilites, but you still have to go through the exercise of the what ifs to figure out what numbers to multiply.

Suppose you want to know how many possible two card hands there are:

The answer is C(52,2) which is read 52 choose 2. This is equivalent to (52!) / ((50!)*(2!)) which is equal to 52*51 / 2 which is equal to 1326. You can also type "52 choose 2" into google instead of using the formula C(n,k) = n! / (n-k)! k!

If that is what you mean by formulas, you can get a lot of that from a basic combinatorics book. That answers questions like how many different ways are there to put different sets of different numbers of things together. But you still have to do the work everytime of figuring out what things you are counting and how to apply the math. There aren't any shortcuts other than trying to figure out common situations and calculate those away from the table. You'll find plenty of info on the odds for common situations.

Or you can use a program like pokerstove.

I guess I'm wondering if someone could walk me thru the math to determine when you have enough equity. I'm assuming you would not make the same raise if it was heads up because there is a small pot (1.5BB) and you are less than 50% to make your hand. Is there a simple formula? i.e. x/y-1=z to calculate pot odds.

Thanks

[tor]

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I guess I'm wondering if someone could walk me thru the math to determine when you have enough equity.

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Raise when equity > 1/# of current players.


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Is there a simple formula? i.e. x/y-1=z to calculate pot odds.

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Pot Odds = bet/#bets in the pot

[Fantam]

There have already been a few posts about pot odds and equity shares so I will try to make this as straightforward as I can.

Lets say you have a hand on the flop like a flush draw. (You have 2 suited cards which match 2 suited cards on the flop).

The chance of your making a flush on the turn is 9/47 (9 remaining flush cards divided by the number of unseen cards) = approx. 19% = approx 4:1 expressed as odds.

OK same analysis for making your flush on the river if you dont make it on the turn. 9/46 = approx. 20% = 4:1 odds.

OK chance of making your flush by the river, when you have your flush draw on the flop with 2 cards to come. (The maths is 9/47 + (38/47 * 9/46) or approx. 20% + (80% * 20%) = approx. 35% = approx. 3:1 odds.

So what does this all mean? I find it easiest to look at this in terms of odds.

If you have a flush draw on the flop your odds of making your flush by the river is ~ 3:1. That means for each bet you put in the pot you would like at least 3 other players to call so that you will be getting value for your bet.

If you have a flush draw on the turn, your odds of making your flush are now 4:1, so you need at least 4 other players to call your turn bet to get value for it.

How does this compare to pot size? You use pot size to determine whether it is worth calling a bet.

So if you are heads up on the turn and have a weak flush draw and the pot size is only 3 big bets, it is usually not worth your calling as you are only receiving pot odds of 3:1 to call compared to at least the 4:1 odds that you need.

Putting this all together it means that usually you should at least call with your flush draws (and open end straight draws) to the river. Thats because you will nearly always be getting sufficient pot odds to make staying in the hand correct.

In practice it is often correct to bet these strong draws on the flop, when you have between a 30% to 35% of making your hand by the river, and depending on the number of your opponents remaining in the hand (say 3 or more), may only have to contribute 25% or less of the bets going into the pot. This is what is known as your having a pot equity edge.

On the turn, when you often have fewer remaining opponents if you are still drawing to your hand, it then usually becomes correct to just check or call. Thats because you usually will no longer have a pot equity edge.

Hope that this makes sense. [img]/images/graemlins/wink.gif[/img]

[sirana]

</font><blockquote><font class="small">In Antwort auf:</font><hr />
Raise when equity &gt; 1/# of current players.

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If I decide to valuebet/raise a draw, how much people do you have to expect to call if you had e.g. 9 outs? Would it be the same as the break-even pot odds, so you would need 4.1 callers (if you leave the raising for free card out)?

[glean]

for just the pot odds that bet would be =ev, in other words if you gained no other benifit from it, free card...folding out stronger hands...disguizing your draw...meta game considerations, you would need just a hair more to bet for value. But you sometimes get other value, and don't forget implyed odds as well. You also halfto consider making your hand and looseing if its not the nuts or if it could be counterfit.

[tor]

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Raise when equity &gt; 1/# of current players.

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If I decide to valuebet/raise a draw, how much people do you have to expect to call if you had e.g. 9 outs? Would it be the same as the break-even pot odds, so you would need 4.1 callers (if you leave the raising for free card out)?

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If you're drawing to nuts, and aren't afraid of redraws then your equity is *exactly* the same as your odds to hit your hand, not the pot odds. If you had the nut flush draw with one card to come and you're about to see the turn card and aren't afraid of redraws, your equity is about 33%. By the rule I stated above, you're good to reaise here against 2 or more opponents.

[glean]

omg, don't belive I made that mistake. tor is obviously correct. its the odds of hitting the hand and not pot odds, which just happen to be about 4:1, same as said pot odds.

Thank you Fantam I think that may be the most understandable explination I have heard yet.

The lightbulb over my head just turned on!

[tor]

The odds here for hitting your hand, for the purposes of calculating equity are actually around 2:1 because you are looking at whether or not you'll have the best hand by the river, not just the next card.

Someone kindly let me know if I'm wrong.