Math; It's Not Just for Eighth Graders

Something I hear very often from people is that I must be very good a math to be good at poker. This is half true, in that you have to be "not terrible" at math to be good a poker. All the math you need, however, was taught to you in eighth grade, and my 5 years at MIT were really not necessary (although 6.041 did help quite a bit....I got an A+ in that class). But you do need to be really good at that eighth grade level math. This hand happened recently in The Oaks 30/60 game, and is a pretty good (and rare) example where you can basically figure out exactly where you stand in a hand and use arithmetic to decide how to proceed.

I find myself in the big blind with 64 offsuit. There is a limper and then a raise, followed by an immediate cold-call from Steven. Steven is one of the better players in the game, but he still has some flaws, for example cold-calling raises first in or in the small blind with very speculative hands. Another player calls the raise, the small blind calls, I opt to call (playing any two cards here at 11:1, basically closing the action with only the very, very slight chance of the first limper re-raising, would be defensible) and the limper only calls. We see the flop 6 ways for two bets a piece. The board comes out:

Q94r

I have flopped a very clean pair. When you flop bottom pair there are a ton of factors that go into whether or not you can proceed. Obviously the number of players left to act, and their likelihood of raising, is important. But board texture also is very important. Is there a flush draw? If so, would any of your "outs" complete a possible flush? What about a straight draw? Would making two pair put a straight on board? Are you likely to be dominated? That is to say, if you make two pair or trips, is it possible that someone will make (or already hold) either a better two pair or have trips with a better kicker? All of these questions are answered correctly (the way that helps me) in this hand. I have bottom pair, it's unlikely anyone else has it, and if I hit a 3rd 4 or a 6 for two pair, I'm probably going to have the nuts. The flop action goes:

SB checks, I check, the limper checks, the preflop raiser checks (this is super important obviously), and Steven pauses and bets. The cold-caller folds, the small blind folds and I face my decision. Getting 13:1 calling is pretty much automatic. Only two players are left to act and both have already checked. The preflop raiser check/raising would be an extreme rarity, and the other guy...well there's a chance but paying two bets wouldn't be the end of the world. So I call, the limper folds and the preflop raiser calls (his hand is basically face up as AK here). For those of you keeping score there are 15 small bets in the pot, we're 3-handed, and the turn brings:

Q94-2r

All four suits are now up on the board, and I still have a 5 outs to make what rates to be the best hand. I check, the preflop raiser checks and clearly doesn't like his hand, and Steven bets. I run through the math quickly in my head and decide I have a very thin call, so long as the preflop raiser isn't making a huge slow play. I glance left and his cards are almost in the muck already, so I call, getting 8.5 : 1 and praying for another pair but planning to fold unimproved but to execute a donk of death if I do spike some help. Alas the river bricks off and I check. Steven checks behind and I declare "a pair of fours." He rolls 98s and drags a nearly $600 pot.

So why is this hand interesting? It's interesting because the turn decision is razor thin and because I had nearly perfect information about what my opponent held and what I needed to win.

First, Steven's holding:

There is almost no draw available on this flop, other than a straight draw with Jack-Ten. He can't have two pair, as he wouldn't call with any of those hands preflop, and he can't have a set of 9s or Queens, as he'd 3-bet those hands preflop. So unless he has the last two fours, which is extremely unlikely, he has exactly one pair (either 9s or Queens) and all my outs are clean. He also can't have Q6 or 96, from the preflop action. Basically if I hit, I win.

Next, my equity:

Against his actual hand, or any other one pair hand that doesn't also have a 6 or a 4 in it, my equity is 11.36%. 5 cards will give me the winner and the other 39 will send the pot to Steven.

Then, the pot size:

Once Steven bets the turn, there are (8.5*60) - 6 = 504 dollars in the pot (the subtracted 6 is for the rake and a tip). I can either fold or call.

If I call:

There will be 564 dollars in the pot and my equity will be .1136*564 = 64.09.

The only fly in the ointment is the slight possibility that he holds a set of 4s (which is only a single possible hand combination against several dozen suited 9s and Qs), which is more than offset by my positive implied odds (if I hit, I plan to donk, and he's going to pay me off some of the time). But ignoring the implied odds and the set of 4s possibility, folding would be a $4.09 mistake. Small, no doubt, but I was quite pleased with myself for not making it.