It's a Cooler: Aces Full vs.....Aces Full

Last night at a NL home game I was in a hand with one other player...definitely a cooler. In poker, the term "cooler" refers to a hand that is very difficult for one player to get away from, since that player quite reasonably believes they have the best hand, but where the other player is more sure or knows they have the best hand, maybe even the nuts. This is a hand that players can get busted on certainly, and at the very least lose a big portion of their stack. Some quick examples off the top of my head in my experience include:

• Flush over flush on the flop, one player is pushing their baby flush
• Any other flush over flush, especially on a board with straight draws and no pairs
• Straight over straight, on a board with no flush and no pairs
• Full house versus full house

In the hand last night, I made a standard raise in late position with AQ off, dealer calls, blinds fold. Flop comes out A-Q-8 rainbow. I check, button bets, I call. Turn is a blank, I check, button bets, I raise, button calls. River is an Ace, based on chip stacks and pot size, I go all-in, button calls. Button shows A-8 for AAA88 and I'm sitting with AAAQQ.

In this scenario, the button did suspect something or was still slow playing their two pair on the turn, since they only called my raise on the turn, I could easily have something like Q-Q for a set or even A-K in that spot. When the case ace fell on the river, there is only one hand that beats the button's hand: A-Q.

Math for this scenario:

a) Odds to get a flop with A-Q-8 given two known starting hands of A-Q and A-8: ((2/48 X 3/47 X 3/46) X 2) + ((3/48 X 2/47 X 3/46) X 2) + ((3/48 X 3/47 X 2/46) X 2) = 0.0003469 X 3 = 0.0010407 or 0.104% or about 1 in 1,000

b) Odds for the case ace to fall on the river = 1/44 = 0.0227 or 2.27%

c) Odds for this board = (a) X (b) = 0.00104 X 0.0227 = 0.0000236 or 0.00236% or about 2 in 100,000

Given the board at the river, what are the odds the button thinks I could have A-Q?

This is roughly 1/45 X 2/44 = 0.001 X 2 = 0.002 or 0.2% or 2 in 1,000. Definitely a cooler.

With this action, and with the chip stacks and pot size on the river, I could easily be pushing trip aces with K-Q kickers (holding AK), or queens full of aces (holding Q-Q), and of course, aces full of queens (holding A-Q). So, theoretically, button could easily believe they are good 2 of 3 times and have to call with their aces full of eights. Again, definitely a cooler.