I see this often, thankfully from my opponents. I like asking students questions like “How often will you flop a set with 77?” Drawing upon their dismal recent history of flopping sets they answer, “About one in twenty times.” This is a pretty important statistic to misremember, and it will massively color how you play that hand.
There are two kinds of math at the table, facts that you recall and those numbers that you calculate. Statistics like flopping a set (1 in 8 times) or flopping a pair with unpaired cards (30% of the time) must simply be memorized. The other kind of math at the table is calculations, I will argue those are easier because one procedure will cover a large number of situations.
The most famous and useful calculations in Hold’em revolve around the Rule of Two and Rule of Four. I have recently derived some add-on’s to these core rules that allow me to estimate equities within a few percentage easily at the tables.
Let’s go through the mental dialog where this math saved me from making a bad call in a tempting spot.
We are five to the flop there is $100 in the pot and we are on the button with a lovely suited Ace. The flop comes down Jack of Flush, Ten of Flush and Deuce of brick. We have the nut flush draw in a multi-way pot.
We just happen to remember that Flushes come in 20% of the time on each card or about 40% by the river. We usually just remember this because over the years we have frequently thought “Nine flush outs multiplied by two is about 20% on the next card.”
We like our spot here.
There is a check and someone bets full pot, then someone else min-raises to $200. This second player is very straight-forward so he has a big hand. We only have $350 back. Remembering statistics will no longer help us in this unique situation, we need to do some math along with our poker feel.
Flush draws get a lot of their value from fold equity. Do we have any of that? We only have the ability to put $150 on top. The second player just min-raised to $200, we doubt $150 more is going to ever fold him out. We just lost one of the most potent weapons of the flush draw, now we are going to have to show Villain a flush to get this pot.
We normally can just say “We get a flush 40% of the time by the River.” But we have some special information here. Using our poker feel, we suspect we are up against top pair from the first player and either top two pair or a set from the second player. We are plenty happy for the top pair player to be in the pot, but the Two Pair or set is a real problem. We can make our flush and easily lose to the redraw.
How often do we win against these specified hands? The memorized statistics are not going to help here, we need another procedure. Rule of Four is not enough to account for these redraws and they are critical.
In my new book, Poker Work Book for Math Geeks, I introduce some very useful additions to the Rule of Four to account for the redraw. Without explanation of the modification, let’s try it.
All of this money is eventually getting in, so really we are getting two cards for the cost of the rest of our stack. Nine outs times four is 36%. Villain very likely has Two Pair or a set. The redraw requires us to give back 20% of our total equity. This means we take a 7% loss of equity. This puts us at 29% equity. Flopzilla says it is really 29.8%. This is a fine estimate for what we need to do.
We expect to get back 30% of the final pot. Under the best case scenario all three of us put money in and we basically break even. Realistically, many times the original raiser will fold and then we are a huge dog.
As much as we love nut flush draws, without fold equity and with Villain’s likely redraw, we are at best breaking even and often going in as a big dog. There is no way to make money, we are going to fold here.