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Part 2 · Lesson 8 of 25 · Free

Your hand against a distribution

Course by Hernan Alejandro Gajardo Morales

KK against a Rock's all-in. He has aces 30% of the time, and against them you win only 18%. You are still a 58% favorite. Against a range, your equity is the average of your equity against each hand in it, weighted by combinations.

The hand

The Rock moves all in for 100 bb into a 25 bb pot. He does this only with AA, QQ and AK. Calling 100 bb makes the pot 225 bb, so the call needs 100 ÷ 225 = 44.4%.

Predict first

He has AA 6 ways, QQ 6 ways and AK 8 ways. Is your KK ahead of his range or behind it?

  1. Ahead: about 58%
  2. Behind: aces crush you
  3. Exactly even: AA and QQ cancel out
Show the answer

Answer: Ahead: about 58%

Weigh each hand by its combinations: (6 × 18% + 6 × 82% + 8 × 69%) ÷ 20 ≈ 57.6%. The aces hurt, but AK is 8 of the 20 combinations and KK wins 69% against it.

Working it out

Your two kings leave him 20 combinations: AA 6, QQ 6 and AK 8, since only two kings are left for his AK. KK wins 18% against AA, 82% against QQ and 69% against AK. Weighted: (6 × 0.18 + 6 × 0.82 + 8 × 0.69) ÷ 20 = 11.52 ÷ 20 = 57.6%, well above the 44.4% the call needs. Without the AK, the same KK would be at 50%.

As a range tightens, the hands that dominate yours make up more of it and your equity falls; against a wider range the same hand becomes a clear call. The lab and the range drills do the averaging for you; this lesson is about knowing what is averaged.

Formula

Equity vs range = Σ combos(h) × equity vs h ÷ Σ combos(h)

Check your understanding

  1. Against a range, your equity is:

    1. The average of your equity against each hand in it, weighted by combinations
    2. The simple average over the hand types in it
    3. Your equity against the hand he most likely holds
    Show the answer

    Answer: The average of your equity against each hand in it, weighted by combinations

    Every combination is equally likely before any action, so combinations are the weights. A simple average over hand types would count AK, with 8 combinations, the same as AA, with 6.

  2. His range: 6 combinations of AA (you win 18%) and 8 of AK (you win 70%). Your equity, in %?

    Show the answer

    Answer: 47.7%

    (6 × 0.18 + 8 × 0.70) ÷ 14 = 6.68 ÷ 14 ≈ 47.7%.

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This lesson builds on ideas from chapter 8 of The Mathematics of Poker by Bill Chen and Jerrod Ankenman (ConJelCo, 2006), explained in our own words; nothing is copied from the book. Poker Math Coach is not affiliated with or endorsed by the authors or publisher.