Poker homework: what hands beat you?

So my coach and I were talking about some different ways of doing board analysis, and he suggested I compute "the hands that beat me" on every street.  Getting significantly better at board analysis is important as I move up limits from the microstakes I've been playing.  While I'll continue to use poker bonuses on different online poker sites to pad my winnings, no bonus is big enough to keep you from being a consistently losing player.

The basic theory I'm following for hand analysis is this:

1. There are 1,326 possible Hold'em hands. 
2. On every street, it's possible to compute the number of hands that beat me and that I beat.
3. Based upon how you improve (and how many hands that used to beat you improve) it's possible to make some intelligent decisions about how to play hands that fall in that weird area that aren't the nuts, but also aren't weak enough to check-call.

Here's the hand I chose to analyze:

Full Tilt Limit Hold'em, $0.50 BB (9 handed) – Full-Tilt Converter Tool from FlopTurnRiver.com

Preflop: Hero is MP2 with A, 3
2 folds, Hero raises, 2 folds, Button 3-bets, 2 folds, Hero calls

Flop: (7.4 SB) 4, 10, Q (2 players)
Hero bets, Button raises, Hero calls

Turn: (5.7 BB) A (2 players)
Hero bets, Button raises, Hero calls

River: (9.7 BB) 6 (2 players)
Hero bets, Button raises, Hero calls

Total pot: $6.85 (13.7 BB) | Rake: $0.30

So it isn't that hard to see why I lost.  By ignoring the hand range of a preflop 3 bettor, I should have easily saved myself at least one big bet (BB) on the river, and possibly one on the turn as well.  But another way of looking at the hand is by examining what hands I could beat that were likely to be playing.  This analysis is another way of seeing that I should have been slowing way down with this hand.  Let's go through it:

Preflop: Hero is MP2 with A, 3
2 folds, Hero raises, 2 folds, Button 3-bets, 2 folds, Hero calls

Flop: (7.4 SB) 4, 10, Q (2 players)

At this point in the hand I've got a nut draw, but actually only Ace High.  Against a random set of playable hands I can beat A2 and KJ.  All the pairs beat me (22+), all the Ace-anything with 4 kicker or better (A4+) along with all the broadways containing a T or a Q (Tx+ and Qx+).  This flop is eminently checkable.  It also shows why if I was holding AK, I could bet this flop.  Because I can beat anyone who missed the board and for many who hit the board, I could out draw them.

If I'd done the "what would a solid preflop three bettor have?" I could have narrowed his range to AK, AQ, and 88+.

Of course I don't do that, I bet, get raised, and call.

I accurately read my opponent here for a Q but figure I can draw out with the Ace or the diamond flush.

Turn: (5.7 BB) A (2 players)

This changes things, of course.  All I see is the Ace, but though I am brought an Ace, that Ace helps a lot of other hands that can beat me.  I can beat KK,JJ,99,88,77,66,55,33,22, but I can't beat Ace-anything except A2.  I can beat Ten-anything unpaired and Queen-anything unpaired, which is what I thought my opponent had.  Had I been analyzing better, I would have seen there's little left that would raise me.

River: (9.7 BB) 6 (2 players)

Finally on the river, my flush doesn't come.  This would be an opportune time to check.  Luckily I did not.  I can beat only Ace-Two, pocket pairs that didn't hit the board, and overplayed queen-something or ten-something.  Everything else hit the board better than me.

Ah, but I bet anyway, and predictably got raised.  My opponent shows down Ace-Queen, no surprise.

It's a good technique for analyzing where you are in a hand, and I hope to do more of it (and a more mathematical version update to this soon.)

Incidently if you're hunting for a new online poker site, make sure you read one or two poker reviews before picking one.  Or you can just find a poker site toplist from someone you trust.