Today, I received an email with some questions about material in *Killer Poker by the Numbers* (*KPBTN*) having to do with probabilities and combinations. I figured I’d answer them here to benefit everybody out there who may have similar questions. Enjoy!

QUESTION #1: Where do the numbers in Table 3.5 on p.84 come from?

Table 3.4 in *KPBTN* deals with a hand where the Hero holds JJ against a single opponent who’s on {JJ-22, AK-AJ, KQ}. Table 3.5 deals with the subset of the opponent’s distribution against which JJ is favored to win, {JJ-44, 22, KQ}, breaking it down with respect to combinations.

JJ: Since Hero has two jacks in his hand, there are only two jacks left in the deck. Therefore, there’s only 1 possible JJ combination.

{TT-44, 22}: For each of these pocket pairs, all four cards of the rank are available. Take TT as an example. There are 4 tens possible for the first card and 3 tens possible for the second card given that the first card is a T. 4*3 = 12. However, that’s the number of permutations…we need to divide out the double counts. There are two ways to have a particular pair of tens (e.g. TcTd and TdTc); therefore, divide 12 by 2 to get 6 combinations of TT. (Alternatively, you can just list every combination and count to see that there are 6: TcTd, TcTh, TcTs, TdTh, TdTs, ThTs).

KQ: There are 4 Ks and 4 Qs available. 4*4 = 16 combinations.

In total, there are 65 total combinations that Hero beats if opponent is on {JJ-22, AK-AJ, KQ}. Meanwhile, breaking down the hands that beat Hero:

AK: 2 As and 4 Ks means 2*4 = 8 combinations.

AQ: 2 As and 4 Ks means 2*4 = 8 combinations.

AJ: 2 As and 2 Js means 2*2 = 4 combinations.

33: 3*2 = 6 permutations. Divide by 2 to get 3 combinations.

In total, there are 23 possible combinations that beat Hero’s JJ.

P(Hero is ahead on flop) = 65/88.

P(Hero is behind on flop) = 23/88.

QUESTION #2: Where does equation 1.11 on p. 27 come from?

Equation 1.11 is the probability of flopping at least one A or K given that you hold AK. To find this probability, find the number of flops with at least 1 A or K and divide by the total number of possible flops.

Number of flops with 1 A or K: There are 6 possibilities for the first card. Given that the first card is an A or a K, there are 44 possibilities (cards left in the deck that aren’t As or Ks) for the second card and 43 possibilities (cards left in the deck that aren’t As or Ks) for the third card. To eliminate the double counts for the second and third cards, divide by 2. In total, there are (6)(44)(43)/2 = 5,676 flops that contain 1 A or K.

Number of flops with 2 As or Ks: There are 6 possibilities for the first card. Given that the first card is an A or a K, there are 5 possibilities (cards that are A or K) for the second card and 44 possibilities (cards left in the deck that aren’t As or Ks) for the third card. To eliminate the double counts for the first and second cards, divide by 2. In total, there are (6)(5)(44)/2 = 660 flops that contain 2 As or Ks.

Number of flops with 3 As or Ks: There are 6 possibilities for the first card. Given that the first card is an A or a K, there are 5 possibilities (cards that are A or K) for the second card and 4 possibilities (cards left in the deck that aren’t As or Ks) for the third card. To eliminate the double counts for the first, second, and third cards, divide by 6 since there are 3*2*1 =6 ways to arrange three objects. In total, there are (6)(5)(4)/6 = 20 flops that contain 3 As or Ks.

In total, there are 5,676 + 660 + 20 = 6,356 flops containing 1 or more As or Ks. Meanwhile, there are (50)(49)(48)/6 = 19,600 total possible flops. (50 possibilities for 1st card, 49 possibilities for 2nd card, and 48 possibilities for 3rd card…divide by 6 to kill double counts with three cards). Probability of flopping at least 1 A or K is 6,356/19,600, which is about .32.

If you need more help with calculations like these, check out chapter 2 in *Tournament Killer Poker by the Numbers*, which is an improved version of the material in chapter 1 of *Killer Poker by the Numbers*. Also, after a few minutes of Googling, I found the tutorial located here. It’s not the best treatment of probability and combinatorics that I’ve ever seen, but it’s not bad…especially for those without really strong math backgrounds. Let me know if you find a free online tutorial that you like better.

May Your EV Always Be Positive!

Tony Guerrera (The Tsunami)

Tags: combinitorics, math, permutations, probability