Micro Limit Math Thread

[sirana]

Ok Tor, I'll give it a try, but don't expect to much (i.e. something correct ;-)

I'll assume a single opponent.
I'll assume the opponents holding are completely random.
(e.g. a very passive BB)
I'll assume you don't hold any of the cards that are of interest.

1. Any two hearts with 3 hearts known
- 10/45*9/44 = 4.5%
(1b. Any two hearts with 5 hearts known
- 8/45*7/45 = 2,8%)
2. Any heart with 4 hearts known
9/45+9/45 = 40%
3. Any Ace or King with an Ace known
7/45 + 7/45 = 31%
4. Any Ace or King with 2 Aces and a King known
- 5/45+5/45 = 22%
5. Two Aces with no Ace known
- 4/45*3/44 = 0,6%
6. Two Aces with one Ace known
- 3/45*2/44 = 0,3%
(6b. Two Aces with 2 Aces known
- 2/45*1/44 = 0,1%)
7. Any Ace with one Ace known
- 3/45+3/45 = 13 %
8. Any Ace with two Aces known
- 2/45+2/45 = 9%

Unfortunally this doesn't help much even against a single opponent since the cards he is holding aren't random. He is more likely to play suited cards than offsuit cards and much more likely to play for example a hand that contains an ace than a hand that contains a two.

I'm really not sure if the math I'm doing here is right, so corrections would be really welcome.

[sirana]

OK. I made some stupid mistakes here, so I try again cause I unfortunally can't edit my post. hope it's right this time

1. Any two hearts with 3 hearts known
- 10/45*9/44 = 4.5%
(1b. Any two hearts with 5 hearts known
- 8/45*7/45 = 2,8%)
2. Any heart with 4 hearts known
- 1-36/45*35/44= 36,36%
3. Any Ace or King with an Ace known
1- 38/45*37/44 = 1- 0,71 = 29%
4. Any Ace or King with 2 Aces and a King known
1- 40/45*39/44 = 21,2%
5. Two Aces with no Ace known
- 4/45*3/44 = 0,6%
6. Two Aces with one Ace known
- 3/45*2/44 = 0,3%
(6b. Two Aces with 2 Aces known
- 2/45*1/44 = 0,1%)
7. Any Ace with one Ace known
1 - 42/45*41/44 = 13%
8. Any Ace with two Aces known
1 - 43/45*42/44 = 8,8%

[sirana]

I've tried a different mathematical approach to tor's question. I'd like to know what is thought about it.

I'm assuming an opponent who uses the tight games chart in SSHE and has limped or raised in EP => his possible holdings are AA-77, any two suited cards T or higher), AK-AJ, KQ)

so out of the 52*51= 2.652 starting hands he only plays 12*8 (the pocket pairs)+ 20*4 (the suited cards T or higher) + 24*4 (AK-AJ, KQ) = 272 possible holdings. Each of this holdings is equally likely with the probability of 1/2652, if we don't know any other cards (so out of this position he plays or raises 272/2652 = one out of ten hands).
But if we do know other cards it gets tricky.

I have written down the holdings arranged by first card to make it a bit easier

A=3A+4K+4Q+4J+1T = 16*4 = 64
K=4A+3K+4Q+1J+1T = 13*4 = 52
Q=4A+4K+3Q+1J+1T = 13*4 = 52
J=4A+1K+1Q+3J+1T = 10*4 = 40
T=1A+1K+1Q+1J+3T = 7*4 = 28
9=3*4 = 12
8=3*4 = 12
7=3*4 = 12

= 272 hands


1. Two hearts with 3 hearts known
a) If none of the 3 known hearts is over T: 20/272 = 7,4% %
b) If one of the 3 known hearts is over T: 12/272 = 4,4 %
c) If two of the 3 known hearts are over T: 6/272 = 2,2%
d) if all three of the 3 known hearts are over T: 2/272=
0,7%

2. Any heart with 4 hearts known

Number of holdings that contain one or more hearts: 20 (2 suited hearts) + 32 (pocket pairs containing a heart) + 48 (Offsuit holdings containing a heart) = 100


(I've only did 3 examples here)

a) If none of the 4 hearts is over 6: 100/272 = 36%
b) If the hearts on the board are K,Q, 3, 6:
24 (pocket pairs) + 6 (suited hearts) + 12
Offsuit holdings containing a heart): 42/272 = 15,4%
c) If the Hearts on the board are A,K,Q,J: 16/272 = 6%

3.) Two Aces with none known: 12/272 = 4,4%
4.) Two Aces with one Ace known: 6/272 = 2,2%
5.) Two Aces with two Aces known: 2/272 = 0,74%

6.) Any Ace with zero known: 116/272 = 42,6%
7.) Any Ace with one known: 84/272 = 30,9%
8.) Any Ace with two known: 54/272 = 19,9%

Now with the any Ace or any K question this becomes really tricky and I'm pretty much at the end of my math. if anybody had an idee how to best solve this, I'd be gratefull.

[Fantam]

Sirana made a good attempt at the calculations, but I think there were possibly still a few errors.

Anyway these are my calculations to your questions, which I hope will be fairly easy to follow:

Any 2 [img]/images/graemlins/heart.gif[/img] with 3 [img]/images/graemlins/heart.gif[/img] on a full board
10/45 * 9/44 = 4.5%

Any [img]/images/graemlins/heart.gif[/img] with 4 [img]/images/graemlins/heart.gif[/img] on a full board
(9/45 * 36/44) * 2 = 32.7%

Any A and any K with an A on a full board
(3/45 * 4/44) * 2 = 1.2%

Any A and any K with 2 A's and a K on a full board
(2/45 * 3/44) * 2 = 0.6%

2 A's with no A's on a full board
4/45 * 3/44 = 0.6%

2 A's with one A on a full board
3/45 * 2/44 = 0.3%

Any other A with one A on a full board
(3/45 * 42/44) * 2 = 12.7%

Any other A with 2 A's on a full board
(2/45 * 43/44) * 2 = 8.7%

I hope this helps. [img]/images/graemlins/wink.gif[/img]

I'm new to the forum and I was hoping someone could give me a better/different explination of equity as it applies to raising. I feel I have a good grasp on pot odds and implied odds, but I almost never raise when I'm on a draw.

For example lets say I'm playing Ax [img]/images/graemlins/club.gif[/img] from the CO. 4 players see an unraised flop and it comes K [img]/images/graemlins/club.gif[/img] 4 [img]/images/graemlins/club.gif[/img] 7 [img]/images/graemlins/diamond.gif[/img] SB bets, BB calls should I call or raise with the button to act behind me and a draw to the nut flush?
What if I'm on the button in the same situation?
In MP?
In a blind?
and how do you do the math to make this estimate?


P.S. I'm too lazy to spell check sorry.

[glean]

I would raise it. You have almost a 50% chance of winning, tho sb could theoreticly be on a set and have a chance of making full, and there is a chance your A could be clean and win the hand for you. Anyhow, with 4 players your getting 3x the money you put in but winning half the time. The other aspect is that someone with a flush draw will probly continue anyway, and if your draw hits they may not put you on a flush and you gain an extra bet or two. Also you may buy the buton incase you hit an A, giving you more info on weather to continue or not since you can close the action. On top of all that you may get a free card on the turn if you don't improve, and with only a 20% chance there you will probably want to take it since, assuming all are still in exept maybe the button since he was faced with two cold, you will be getting less money in for your money than your chance of improving.

Hope that made some sence, tho I have a feeling it isn't the most consice way to explain it at all.

[tor]

I appreciate the math you guys did. Though, it's not what I was looking for. I suspect the problem lies in that I used the word "frequency" in my title, but in the post was asking about total number of combinations there are in making a specific hand. I.E. I hold A[img]/images/graemlins/club.gif[/img]K[img]/images/graemlins/heart.gif[/img] There's K[img]/images/graemlins/spade.gif[/img], J[img]/images/graemlins/heart.gif[/img], 2[img]/images/graemlins/heart.gif[/img], 7[img]/images/graemlins/heart.gif[/img] Q[img]/images/graemlins/heart.gif[/img] on the board, how many ways can he hold an A[img]/images/graemlins/heart.gif[/img] and a pair.

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.

[Fantam]

I think I see what you are asking.

You are trying to determine the different number of possible combinations for say a A [img]/images/graemlins/heart.gif[/img] and a pair in your example, without counting A [img]/images/graemlins/heart.gif[/img]K [img]/images/graemlins/diamond.gif[/img] etc in your head.

Well how about saying that his first card has to be the A [img]/images/graemlins/heart.gif[/img] and then the 2nd card must pair the board. Then the 2nd card could be 2K's,3J's,3 2's,3 7's or 3Q's.

So there are 2+3+3+3+3 different possible 2nd cards.

Or 2+(3*4) different possible combinations = 14.

With only a little bit of practice you ought to be able to work out the different number of possible combinations quite quickly.

[Dave G.]

I don't think you're going to find a set of formulae simple enough to use at the table for these situations unless you are really good at doing arithmetic in your head (in which case you'd just be using standard probability equations and the conditional probability equation).

How I do it is I just memorise some standard results and use that as a starting point for a situation I might come across. I adjust the number in my head using my judgement of how the other factors at play might affect this number. There's no way I'm doing these calculations while playing - if I had to do this for each of the 4 tables I play on every hand I play, I'd only be able to play for a short time before I became fatigued from all the concentration.

The correct middle seat 3 (left of button in this case) play here is to raise. Unless button has a monster hand, its highly likely he will fold. If he has a good draw and calls, your is better...
When you raise, 2 things will happen on the turn,
1. u hit and ace or club...continue betting.
2. you miss it and get a free card (check)

People at any decent level will recgonize what has happened here and often not call your river bet. If you are utg, or any blind, you will not be able to buy a free card most of the time so your raise has no value and may get you in a lot of trouble if someone has a good hand like a set.
your flush should come about %35 of the time, and the ace adds another %5 or so assuming your not dominated. Youll draw your hand less than half the time therefore your hand equity isnt good enough to value bet. In these positions, call the single bet, and hope to turn the flush or ace. If you dont hit it, you can continue to fish until the river in which you need to fold if you dont hit. Also, if your draw hits on the river dont plan for a check raise, almost all players recgonize the flush hitting in relationship with your flat calling is a huge threat.

hope i could help,
chris