AKs vs. AKo

[Toro]

Texas Holdem, heads up situation: AK suited vs AK offsuit. Which hand will make the most flushes? Intuitively it seems that the suited hand should fare better but my personal observences, admittedly unscientific, are that I've seen the unsuited hand make more flushes. Always wondered what the true odds were. Anyone know how to calculate this?

[thylacine]

http://www.twodimes.net/poker/

pokenum -h as ks - ah kc
Holdem Hi: 1712304 enumerated boards
cards win %win lose %lose tie %tie EV

As Ks 122556 7.16 37210 2.17 1552538 90.67 0.525

Kc Ah 37210 2.17 122556 7.16 1552538 90.67 0.475

(sorry don't know how to make nice table.)

%tie 90.67

%win As Ks 7.16

%win Kc Ah 2.17

[mr_jmac]

Hey,

For example: A [img]/images/graemlins/heart.gif[/img]K [img]/images/graemlins/heart.gif[/img] vs A [img]/images/graemlins/spade.gif[/img]K [img]/images/graemlins/club.gif[/img]

For the <font color="blue"> suited hand </font> to make a flush there must be either 3 hearts, 4 hearts or 5 hearts on the board. So, we need to calculate how many boards like this are possible.

Of the 48 unseen cards 11 are hearts. So for 3 heart boards we find all of the possible combinations of 3 hearts from the remaining 11 ( C(11,3) ). Each of these combinations can be coupled with 2 of the remaining 37 cards ( C(37,2) ). We do this for 4 and 5 heart boards similarily.

3 heart boards = C(11,3)*C(37,2)
4 heart boards = C(11,4)*C(37,1)
5 heart boards = C(11,5)

Total = 122,562

For the <font color="blue"> unsuited hand </font> , to make a flush there must be either 4 spades, 5 spades, 4 clubs or 5 clubs on the board.

Of the 48 unseen cards, 12 are spades and 12 are clubs.

4 spade boards = C(12,4)*(36,1)
5 spade boards = C(12,5)

The calculations for club boards are identical so we add these terms and double the result.

Total = 37,224

So, you are roughly 3 times more likely to make a flush with the suited hand.

Note: Total number of boards = C(48,5) = 1 712 304

Probability of making flush with suited hand
= 122 562/ 1 712 304 * 100
= 7 %

Probability of making flush with unsuited hand
=37 224/ 1 712 304 * 100
= 2%

Later,
Joe

[Toro]

Very impressive guys. As is usually the case, perception is not reality. Thanks for the responses.

[Lost Wages]

mr_jmac,

A couple of tiny nitpicks. One is that both hands will make a flush when the board is all diamonds. The other is that a few of the hands that you have counted as flushes are in fact straight flushes.

Lost Wages

[mr_jmac]

Thanks ... I knew when I was responding that I would be forgetting something (not that they change the outcome much.

Glad to know someone is reading [img]/images/graemlins/smile.gif[/img]