Card Removal Effects

When playing poker lately (online, that is), I find myself thinking more than in the past about card removal effects.

Let me set the table for this post this way. If you have an even money bet on something where you are a 51% – 49% favorite, it’s a profitable bet (in a math sense, meaning if you can repeat the bet a million times, you will make a profit in the long run). We call this “positive expected value,” or in poker shorthand, +EV. If you are a 49% – 51% underdog, the bet is unprofitable, or -EV.

It’s like elections for public office, where a shift of one percent or even less can change the outcome. In poker, the consequences aren’t as dramatic (and by the way, please VOTE this November!), but small shifts can make the difference between winning and losing.

OK, back to card removal effects.

In Texas Hold’em, the probably of one of your hole cards being an ace is approx. 15.1%. (Here’s the math: there is a 4/52 chance (7.69%) of your first card being an ace. If it isn’t, now there is a 4/51 chance of your second card being an ace (7.84%). Add those together, then subtract the 0.45% chance of both cards being an ace, and you get 15.08%.)

Suppose you are in a heads-up game. You have a 15.1% chance of having an ace. But so does your opponent, right?

Actually, no. If you don’t have an ace, you should calculated your opponent’s odds after the removal of your cards from the deck. Instead of the first card being 7.69% likely to be an ace, now it’s 4/50 or 8.00%. Notice the effect of card removal on the denominator. Finish the math… and now he’ll have an ace 15.67% of the time.

What if you do have an ace? Now there’s only three aces remaining in the deck, so we can reduce both the numerator and denominator, to 3/50 (and 3/49 for the second card when the first one isn’t an ace). He’ll only have an ace 11.88% of the time when you also have one.

Next, imagine a full table of nine players, you are on the button, and everyone in front folds. Consider two scenarios.

SCENARIO 1: You have pocket aces. You raise and both of the blinds fold.

SCENARIO 2: You have middling suited connectors. Let’s say 98s. You raise and one of the blinds re-raises!

Sounds familiar, but still seems unfair, amiright?

In both scenarios, we can assume the players in front of you had a lot of junk. While there may be some Ace-rag or King-rag combinations included in that junk, the folded hands most likely contained a disproportionate number of low cards. That leaves disproportionately more high cards in the deck for you and the blinds.

When you have pocket aces in Scenario 1, the card removal effects on the potential combinations available to the blinds takes away a sizable chunk of their calling ranges. Half of their possible AT+ combos are removed. Compare that to Scenario 2. It may look like you were trying to steal the blinds and one of them responded with a re-steal. Or not. Don’t underestimate the card removal effects, creating what blackjack players would call a hot deck that boosts the probability of one of the blinds waking up with a legitimately strong hand.

Here’s a simple adjustment for card removal effects. Stop raising from the button (after everyone in front has folded) with most suited, connecting, and 1- or 2-gap combos if you are mainly hoping to steal the blinds. Instead, raise with most A-x or K-x hands, no matter how junky looking, as the removal of your A or K forces the blinds into weaker ranges.

That’s a tip you can take to the bank! And be sure to look for other scenarios where card removal effects should be considered before doing the poker math.

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