jjd323’sMathematics of PLO Ep. 1

Basic Starting Hand Combinatorics - AAxx

Combinations

• In Excel: “=COMBIN(n,k)”• By hand: “C(n,k)”• On your calculator: “nCk”• http://en.wikipedia.org/wiki/Combination

PLO vs. NLHE

• In Excel: “=COMBIN(52,4)”• By hand: “C(52,4)”• On your calculator: “52C4”

C(52,4) = 270725

• cf. to Texas Hold’EmC(52,2) = 1326

270725 / 1326 = 2041/6

An example: How many AAxx ?

• What % of hands are AAxx?– Count combinations containing AAxx and divide by

total number of possible starting hands.• Break the problem into two parts –

– how many ways to deal AA– how many xx to go with them?

C(4,2) ways to get AA, leaving 50 other cards C(50,2) xx for each pair of AA

But we get the wrong answer as we count AAAx and AAAA hands multiple times.

How many AAxx ?Naïve method:C(4,2) = 6 ways to get AA, each leaving 50 other cards in the stubC(50,2) = 1225 ways to get xx with each pair of AABut doing this, we count AAAx and AAAA hands multiple times.

e.g. AsAdAc2c is counted once, but then AsAcAd2c is counted again, and this double-counting leads to the total count being too high.

Correct method: C(4,2) = 6 ways to get AA, with 48 non-A cards remainingC(48,2) = 1128 ways to get xx with each pair of AABeing careful to not double-count AAAx and AAAA hands.

C(4,2) * C(48,2) 6 * 1128 = 6768 AAxx hands

How many AAAx and AAAA ?

Counting AAAx handsC(4,3) = 4 ways to get AAA, leaving 48 non-A cards

C(48,1) = 48 ways to deal x with each set of AAA

Counting AAAA C(4,4) = 4 ways to get AAAA

• 6768 AAxx hands• 192 AAAx hands• 1 AAAA hand

6961 total AAxx hands

How many AAxx ?• 6768 AAxx hands

• 192 AAAx hands

• 1 AAAA hand

• There are C(52,4) = 270725 total starting hands in PLO

6768 / 270725 = 2.50% (AAxx only)

6961 / 270725 = 2.57% (any AAxx)

cf. AA in Texas Hold’Em is

6 / 1326 = 0.45%

Other Tools & Info• WiltOnTilt’s Mathematics of NLHE:

– http://www.deucescracked.com/videos/8-Mathematics_of_NL_Holdem/20-Season_Premiere

• You can check this at propokertools beta using the “count” tool:

– http://beta.propokertools.com/simulations

• Use Excel to calculate complex cases

• Sean Poker blog post:

– “Breakdown of an Omaha Preflop Range” http://seanpoker.net/518/articles/breakdown-of-an-omaha-preflop-range

– “Omaha Preflop Range With ProPokerTools” http://seanpoker.net/821/articles/omaha-preflop-range-with-propokertools

• Tom “LearnedFromTV” Chambers PLO book:

– http://plotheory.com/Book/tabid/77/Default.aspx