Hold 'em query - All in with top pair; Odds?

I've subscribed to the WoV forum to field a query on the odd's of winning a hand in hold'em when going all-in pre-flop and holding top pair against another pair - the classic scenario of course is AA v KK. I did wish to ask the Wiz himslef on his WoO site, but the advice was to post the question here.

So . . . in more than one poker book I've read, and on many web sites, the odds of winning with top pair on a two pair pre-flop all-in call is shown as 70%, with a 30% chance of the top pair being outdrawn. I don't think this is correct. Watching a game on TV not too long ago, the same situation came up, and the top pair was shown at 80%. I don't this is correct either.

Here's my reasoning, and I'd be grateful if anyone with greater maths knowledge than myself can set me straight if I have this wrong.

Pre-flop, with a pair in hand, there's a 20% chance of making a set - 2/50 (outs divided by unseen cards) x 5/1 (number of possibilities of the outs hitting the board)= 10/50 or 20%. So there's an 80% chance for either player that they won't see another card of whatever value it is they're holdong.

So now to the coincidences:
The chances that neither player will hit, so top pair wins, is 64% (80% x 80%)
The chances that both players will hit, so top pair wins is 4% (20% x 20%)
The chance that one or other of the players will hit a set is the difference - 32%. As both players have
an equal chance of this, splitting it equally down the middle gives both a 16% possibility of drawing
to a set and winning?

So adding all of these up:
Top pair's chances of winning are - 64% (no sets) + 4% (both hitting a set) + 16% (top pair improving to a set) = 84% in total.
Under pairs' chances are just 16% (where it improves to a set and top pair doesn't).

There is of course the outside possibility that one could draw to a set, and one to a quad, and I've estimated
this assuming the first two cards to hit the flop are both players' set draw cards:

3/48 (quad card hitting 3rd, 4th or 5th slot on the board) x 1/25 (possibility of two sets showing) = 1/400, or 0.25%; effectively
nothing that will materially affect the calculation above.

Can someone look over my calculations please?

Thanks in advance.

You've left out a lot of other possibilities, including bottom pair drawing a straight or flush. If the top pair is lower than aces, then there is the chance of a full house on board of X over aces (or any other pair that beats top pair). Splits due to quads with high kicker, straight or flush on board. Etc.

Also, when you see the odds on TV, depending on the show, they may take the folded hands into consideration when they calculate the odds.

A, A vs K, K - (both Kings are the same suits as the Aces) gives Aces 82.4%, Kings 17%

If the kings are 2 different suits to the Aces then this gives Aces 81%, Kings 18.5%

That's why no-limit is the king of poker games. And you just got your answer from one of the royal players.

Thanks guys. Yeah, there are some other possibilities that effect the odds, such as potential straight draws, but fundimentally the figures given by others align with my own.

SWITCH, was Stephen also a hold'em player ? Damn, what a resume.

Quote: buzzpaff

SWITCH, was Stephen also a hold'em player ? Damn, what a resume.

Yes, Steve was a very good Hold'Em player and also a strong Omaha cash game player. We set up a poker club in Birmingham and played against a lot of excellent Omaha players. In fact, I was with Steve one night after a game, discussing a poker name for Dave Ulliott, when he came up with 'Devil Fish' - and that name sort of stuck :-).

Think he could come up with a name for Dan Lubin . One that could be said in the presence of ladies, that is ?

Quote: buzzpaff

Think he could come up with a name for Dan Lubin . One that could be said in the presence of ladies, that is ?

They are all animal names and are given to poker players.

A crude way to come up with it on the fly is like this:

For every 'out' that your opponent has, give him a 2% chance to hit, multiplied by the number of board cards yet to be shown.

ex: With A-A vs K-K all-in pre-flop, he has 2 outs (the 2 remaining kings) x 2%, x 5 board cards to come = 20%

Of course this doesn't factor in outs that are gained via straight/flush, etc. I find this method handy to come up with rough percentages post-flop.

ex: You hold A-J, he holds 5-6 suited, and the flop has come J-8-7 with 2 of his suit. You lead with top pair, but he has 15 outs (9 remaining of his suit plus the non-suited 4's and 10's) to take the lead via straight or flush. So 15 outs x 2% x 2 cards to come = 60% chance of him improving to take the lead. Note that it is NOT a 60% chance of him having the best hand after the river since you can improve as well.

Like I said, it's fairly crude, but it gives you a rough idea.

' The math is easy. Putting your opponent on a hand , not so easy. LOL
Seriously most of the math can only be successfully applied in limit games, not no-limit as in all-in ?

Not at all, plenty of math in the no limit game. Ask Ferguson...

Yes indeed but it is best used in heads up. And if you use only math, you lose. I was just suggesting he not tilt at windmills.
Poker, horses racing, even sports handicapping is a better use of his time !

We are talking about poker.... Sure math is only part of the equation, (ho ho), but even with bluffs, card reading and moves, you can apply mathematical models to increase your knowledge of the works.