Who wants to figure this out for me?

[Dynasty]

A post in the "Poker Theory" forum in which the merits of KT vs. 76 on a K65 rainbow flop is being debated made me think of something.

We know that AK will flop top pair about 1/3 of the time. In fact, any non-pair hand will flop a pair about 1/3 of the time. However, all hands besides AK won't always flop top pair. For example, When AQ flops a pair of Queens, the pair will not be top pair if a King is on the flop.

So, how often do the big offsuit hands flop top pair? AK? AQ? AJ? AT? KQ? KJ? KT? QJ? QT? JT?

I don't remember every seeing this kind of data in anything I've read. I think it could be useful.

[BruceZ]

[ QUOTE ]
So, how often do the big offsuit hands flop top pair? AK? AQ? AJ? AT? KQ? KJ? KT? QJ? QT? JT?

[/ QUOTE ]

That would be real easy to crank out. Do you just want top pair, or do you want to include 2-pair, trips, full houses, and quads, or some combination of those?

[ QUOTE ]

I don't remember every seeing this kind of data in anything I've read. I think it could be useful.

[/ QUOTE ]

And useful to you means it's worth money. Just how useful are we talking? [img]/images/graemlins/wink.gif[/img]

[Senor Choppy]

The most useful would be top pair alone, since all unpaired hands are going to flop 2 pair, trips, etc. equally, unless you were to also look at how often they flopped top 2.

[BruceZ]

[ QUOTE ]
The most useful would be top pair alone, since all unpaired hands are going to flop 2 pair, trips, etc. equally, unless you were to also look at how often they flopped top 2.

[/ QUOTE ]

I was referring only to hands which paired the top card, but also made 2-pair, trips, full house, or quads. Like 2-pair with the top pair, but not necessarily the top 2 pair. Or top trips, top quads, or full house using the top card on the flop. This would be a different number for different hole cards just as top pair is different.

You bring up a third possibility, which is simply top pair or any hand ranked higher than top pair, even if it doesn't contain top pair. As you say, this would simply add a constant amount to every top pair hand, but we may still want these numbers.

Let me know what you want to include. I have developed a general formula and a spreadsheet which allows you to easily enter any preflop hand, and I can configure it to include any of these variations.

[Dynasty]

Clarkmeister sent me a MP saying a chart in Carson's book has this info. So, nobody do any serious work. If you want to post that chart though... [img]/images/graemlins/smile.gif[/img] .

[Dynasty]

If it truly is easy, I was intersted only in flopping top pair.

[aloiz]

I took a stab...

Assuming that we hold two offsuit cards, and that the only hand we make is top pair (no two pair, trips, or boats).

AKo = 2 * (3 * C(40,2) / C(50,3) - 11 * 6 / C(50,3)) = .283
AQo = .1414 (odds for ace making only top pair)+ [3 * C(40,2) / C(50,3) - 10 * 6 / C(50,3)] = .258
AJo = .1414 + 3 * C(36,2) / C(50,3) - 9 * 6 / C(50,3) = .235
ATo = .1414 + 3 * C(32,2) / C(50,3) - 8 * 6 / C(50,3) = .215

If I'm doing this right, the calculations get pretty repetitive.

KQ = .233
KJ = .21 (exactly)
KT = .190
QJ = .187
QT = .167
JT = .147

aloiz

[Senor Choppy]

Wow, those are pretty eye-opening. AK being almost twice as likely to flop TP than JT is nuts. I knew there was a reason that hand sucked so badly.

[BruceZ]

My general formula is:

[ 3*(44 - 4*N)*(40 - 4*N)*3 + 3*(44 - 4*M)*(40 - 4*M)*3 ] / (50*49*48)

where N and M are the numbers of ranks higher than each of your hole cards which you don't already hold. For example, for KJ, N = 1 for the A, and M = 2 for A,Q. For the first hole card, there are 3 ways to pair it, 44 - 4*N ways to deal the second card so it doesn't pair either hole card and is not an overcard to the pair, and 40 - 4N ways to deal the third card so it doesn't pair either hole card, doesn't pair the board, and is not an overcard to the pair. Then multiply by 3 since the pair can come in any position. This is all repeated for the other hole card.

<font class="small">Code:</font><hr /><pre>
hand overcards (N) overcards (M) top pair

AK 0 0 26.9%
AQ 0 1 24.5%
AJ 0 2 22.3%
AT 0 3 20.3%
KQ 1 1 22.0%
KJ 1 2 19.8%
KT 1 3 17.9%
QJ 2 2 17.6%
QT 2 3 15.7%
JT 3 3 13.7%
</pre><hr />

Note that we can verify AK as 6*44*40*3/(50*49*48) = 26.9%. The rest are the same except for different values of N and M.

I can email you the Excel file that does this if you want.

[MushashiAce]

Hey bruce, thanx for responding to all these % questions, I've been coming across alot of your odds questions lately, real good stuff [img]/images/graemlins/wink.gif[/img]