# Math, Combinatorics and Frequencies

NOTE:  This entry was originally posted on a different site on June 16, 2016 and has been slightly edited prior to re-posting here.

I played this hand at a private cash game a few days ago.  On the river, it was obvious that I needed to fold.  Then again, maybe not.

With QQ in middle position, I raised to 8 BBs following a single limper.  This is a bit more than normal for me, however at this game there were frequently multiple callers pre-flop so I decided to let them pay me a little extra.  Or thin the field.  Either would be fine.

There were 3 callers, making the pot 34 BBs, already a bit bloated.

On a flop of K54 with two spades, I bet 16 BBs and the button called.  For purposes of this blog, I’ll call him “Adam.”  Adam is a 23-year-old loose/aggressive thinking player.  He likes to play mixed games and finds Texas Hold’em a bit boring, but plays it because that’s what’s available around here.  Although I’ve only played with Adam a few times, he is willing to mix it up, gamble and be aggressive if he senses an opportunity to steal a pot.

One the one hand, I’m targeting a hand that will call my slightly less-than-half-pot bet like 66-JJ, A5s, A4s or 56s.  On the other hand, I’m concerned about Adam holding either a King or a flush draw with two spades in his hand.  If he does have a King, it’s probably not AK as he would be more likely to re-raise pre-flop on the button.  But it could be KQ or KJ, maybe as weak as KTs.

The turn is an off-suit deuce.  I bet 20BB more and he calls.  My bet is deliberately small, hoping he will call with weaker pocket pairs or other non-flush draw / non-Kx hands that might fold to a larger bet.  He will definitely call with a flush draw and that’s a risk I’m willing to take.  If he has a King with a strong enough kicker, he might raise just in case I am the one chasing a flush.  I know he’s capable of playing his draws aggressively, but not 100% of the time.

The river is the 8 of spades, which completes the flush (if that’s what he’s chasing).  Now I’m faced with a situation that Bart Hanson at Crush Live Poker calls “5th Street Chicken.”  This is where I’m out-of-position, and don’t want to put any more money in the pot.  But if I check, I’m opening the door for Adam to bluff if he actually has one of the hands I’m targeting.

Quick recap:  The board is Ks 5h 4s – 2d – 8s.  I have QQ.  Adam is on the button and called my pre-flop raise, and called my flop and turn bets.  There is now 106 BBs in the pot.

The pot is really too big for my 1-pair hand.  I don’t even have top pair.  If I bet on this river, am I essentially turning a hand with showdown value into a bluff?  Yes.  Is that a good idea?  No.

I check.  Adam bets 32 BBs.

Clearly I have to fold.  He either has the King, or he hit his flush draw, and he’s betting for value.  His bet is small, which it has to be after I waved a white flag by checking the river, all but announcing that I don’t have a flush, nor a hand that is strong enough to bet/fold (for value).

What would you do?  Leave a comment below…

Math

I decided to fold, but before relinquishing my cards started doing the math.  There are 106 BBs in the pot.  Adding his bet of 32 BBs makes it 138 BBs.  By calling, I’d be risking 32 BBs to win 138 BBs means I’m getting pot odds of 4.3-to-1.  I would have to win 1 out of 5.3 times for calling to be profitable, in the purest poker mathematics sense.  That’s slightly less than 19%, a pretty low threshold.  I was recently reading a limit Hold’em strategy book, and recalled some commentary about calling on the river.  Often you will be getting pot odds of 10-to-1 or more in a limit game due to the constraints on bet sizing.  The author’s point was that while most players should fold much more often pre-flop and on the flop, they should call on the river when they have showdown value and there is any chance they are good as little as 8-10% of the time.  That’s just how the math works.

So I ponder this for another minute.  Adam is capable of turning a weaker hand into a bluff here.  What does he think I have?  I went bet-bet-bet-check.  My range can easily be 99-QQ, AK, KQ, KJ, and is probably pretty transparent at this point that I have a 1-pair type of hand.  After I checked the river, the 3rd spade coming in is a great bluffing card for him.  With a King, he’s more likely to check back after the scare card arrives.  But I think he’s figured out that I can fold when it’s obvious that I’m beat.  And he’s got the stones to take advantage of my discipline and tight image.

Combinatorics

I don’t do the combinatorics at the table, but there are far fewer flush draw combos in his range than other combos.  If I include literally any two spades that include an Ace or have two gaps or fewer, that is about 24 combinations.  Plus 3 combinations of pocket 88’s that binked the river for a total of 27 value combos.  I’ll assume that he always bets with these hands.

His non-flush draw, non-Kx calling range for the flop & turn (66-JJ, A5s, A4s, 65s)  has about 39 combinations.  These are the hands that I beat and was targeting with my small bet-sizing.

Frequencies

Now the questions is how frequently will Adam turn one of these 39 hands that I beat into a bluff on the river?  If the answer is 7 or more out of 39, then the mathematically theoretically correct response for me is to call.  Add 7 bluffs to the 27 value combos, and I win 7-out-of-34 times, or 20.6%.  If I do call his river bet, I’m probably going to lose this pot, but poker theory tells me I’ll lose less over the long run by calling in this spot than by folding.  It feels really messed up to have to think this way.  Nevertheless, I think his bluffing frequency with this set of facts is greater than 7-out-of-39, probably closer to 13-out-of-39 (33.3%) or more.  I feel like I’m about to throw away 32 more BBs because of poker math.

To succeed at no limit Texas Hold’em, however, you have to trust your reads.  My read on Adam says his bluffing frequency is high enough for me to call.  And my read on this situation overall is that the most important question right now is the one about his bluffing frequency.

So I call.

He says “I have a pair of fives.”  With 6-5, the turn card gave him a gutshot straight draw to go along with his weak pair… just enough for my small bet to keep him in.

When he sees my cards, he says he figured it was something like that.  “What do I have to do to get you to fold?”

I start to say “Math” but shrug my shoulders instead.

======

If you like this blog entry, help us get more exposure by liking and sharing with your friends on Facebook, Twitter, etc.

## 1 Comment

1. John Mac says:

Excellent call – I would folded 😜!

Like