Blockers – The math

Blockers

How do blockers affect combos?

You have 2 known cards in your hand – therefore your opponents cannot have these in there hand. These cards could be blockers to key hands in your opponent’s range. This information can be crucial. If you think you know it all, try UnfairPoker.com’s interactive combinatorics and probability quiz, which also has questions about blockers.

Let’s say you have been dealt A♥️K♠️ preflop, and you open UTG on a 9 handed table. A straightforward tight passive villain raises you next to act, and you feel that he would only do this with AA, KK, QQ, and AKs. How many combos does he have of each?

Without any blockers, you would conclude he has:

  • AA = 6 combos
  • KK = 6 combos
  • QQ = 6 combos
  • AK suited = 4 combos
  • (total = 22 combos)

So, for example let’s say instead you had opened JJ UTG and faced the same action. This time you have no blockers to range you are assigning villain, and conclude that 18 out of 22 times (when he holds AA, KK, or QQ), you are crushed (and have around 20% equity). 4 out of 22 times (when he holds AK suited) it’s a coinflip (you have around 50% equity). Combining this information with pot odds, implied odds, SPR etc. you can make a decision about how you will proceed in this hand.

With your A♥️ blockers and K♠️ blocker, his possible combinations change to:

  • AA = 3 combos
  • KK = 3 combos
  • QQ = 6 combos
  • AK suited = 2 combos
  • (total = 14 combos)

It is extremely intuitive how many suited combos are left when you hold blockers. Each hand type can only have 4 suited combos. If you hold 1 blocker, there are 3 suited combos left. If you hold 2 blockers, there are 2 suited combos left. For example, without any blockers there are 4 combos of AKs (A♠️K♠️, A♥️K♥️, A♦️K♦️, A♣️K♣️). The fact you hold the Ah makes the 2nd of these impossible, and the fact you hold the Ks makes the 1st of these impossible – leaving you with just 2 of the original 4 combos. You should note that if you held A♥️K♥️ (or any AK suited) in your hand, this only blocks 1 of 4 suited combos of AK suited.

A quick way of working out combinations, when you hold a blocker is to multiply the relevant unknowns together (if you are working out combinations of a pair, you must divide this answer by 2, because the order doesn’t matter).

AK

4 aces in deck x 4 kings in deck = 16 combos of AK

If you have the Ah and Ks in your hand, there are only

3 aces in deck x 3 kings in deck = 9 combos of AK

AA

(4 aces in deck x 3 aces in deck) / 2 = 6 combos of AA

If you have the Ah and Ks in your hand, there are only

(3 aces in deck x 2 aces in deck) / 2 = 3 combos of AA

He has QQ 42.9% of the time (i.e. 6/14) when you hold A♥️K♠️, compared to 27.2% of the time (i.e. 6/22) when you hold JJ. Comparing your equity with A♥️K♠️ versus each part of his range (which you can weight according to their combos), and combining this information with pot odds, implied odds, SPR etc. again you can make a decision about how you will proceed in this hand.