Probability so. holding a better hand preflop

[peyote]

I'm sure this has been asked before but couldn't find anything...

Given the following scenario:
I'm UTG in a 6-handed Holdem game and I raise the X percent best hands (say 10%)... What is the probability that at least one player holds

- a better hand (vs. my range)?
- a hand from the same range (the X percent best hands)?

Thanks in advance...

[jay_shark]

How would you define a better hand ?

For instance , pocket 2's fair better than ace king heads up if you can see all 5 board cards . But you would rather have ace king in a heads up match than pocket 2's . We can certainly say that AKo is better than AQo but can we say that AKo is better than AQs in a multiway pot ?

An interesting question to ask is the probability that if you hold something like a-10 utg 6 handed that you will face a dominating hand ;10+ aj+ .

[peyote]

[ QUOTE ]
How would you define a better hand?

[/ QUOTE ]

Let's define it as a hand from the same range with a equity >50% against the range... Does this make more sense?

[jay_shark]

First off you can use pokerstove to figure out what hands fair better than your pre-assigned range . Once you've established this range , you can use simple combinatorics to establish how many hand combos there are .

ie you hold A10 utg and your range normally consists of 88+ A9s+ k10s+ QTs+AJo+KQo . This means your opponent should re-raise or at least call with hands like AQ+ 10+ . Lets calculate the number of hands that your opponent would at least call with .

aq: 12 hands since there are 3 aces multiplied by 4 kings
ak: 12 "
10+: 3+6+6+6+3 = 24

So in total there are 12+12+24= 48 hands that are good .
Here is a very good approximation :

Label the players 1,2,3,4,5 . The probability player 1 (or any for that matter ) doesn't hold any of the 48 hands is (50c2-48)/50c2 .Take this number to the power 5 and then take the complement .

1- [(50c2-48)/50c2]^5 = 18.1%

Let me know if you need more clarity .

[jay_shark]

As a side note . You shouldn't be raising with only the top 10% of hands 6 handed . I sure hope you only used this number as an example because otherwise you're playing too tight .

[peyote]

Thank you very much, this helped me a lot...

btw, I raise utg 11.5% hands. Is there any mathematical reason for your statement that 10% is too tight? I don't think this would be that bad in very loose/lag games...

[jay_shark]

This may convince you that you should play more hands . Although this topic can become a bit too subjective for my tastes .

Every time you raise from utg , you have a shot at picking up 1.5 BB's if everyone folds . Lets simplify the game and assume that there is no flop and the best two card hand wins . Assume you raise with the worst hand from the top 10 % range 6 handed ; which is A9s . This means there is an 18.1% chance that someone holds a better hand than yours and will be a considerable favorite and at best you will win around 33% of the time . Here is a quick EV calculation to indicate how you can earn more money by raising with more hands than the ones you've indicated .

EV : 1.5*0.819 +4.5*0.181*1/3 -3*0.181*2/3=1.138 >0

In other words , you should keep raising with more hands from your range such that the minimal hand in your range breaks even from an EV point of view . However , I must specify that you also have to keep in mind that at larger tables when you raise , you will be out of position which may negate the value of your cards over your opponents' cards . Another thing to keep in mind is that most of the value you earn comes from postflop play so how you play your cards on the flop is of utmost importance .