# It’s Always a Denominator Problem

Poor denominators!  They are the red-headed step children of basic math.  Nobody likes them and even worse, nobody understands them.

If you ever see someone struggling with basic math that involves a fraction, decimal, percentage or other ratio of one number to another, you don’t even have to pay close attention to them.  Just tell them to double-check the denominator.  If they still can’t get it right, and you’re feeling sufficiently benevolent, you can ask them what they are using for the denominator, note that it’s wrong, and tell them the correct figure.

It’s always a denominator problem.

A friend called me a few days ago.  For purposes of this blog, I’ll call him “Jeff.”  Jeff’s was staking another player in a poker tournament via the poker crowd funding site YouStake and asked how he should calculate his share of potential winnings.  I guess that makes Jeff an optimist.

Full disclosure:  I’ve pissed away a few of my own dollars backing players on YouStake, with no return on these investments so far.  Yet I’m still an optimist too.

Back to Jeff.  Suppose Jeff had invested \$200 to stake a player in a single tournament with a buy-in of \$5,000.  The player was offering to sell a total of 40% of his action.  What is Jeff’s share of any winnings?

First we need a numerator.  The player Jeff is staking might charge a markup, for example 1.15, and YouStake charges a 8.3% site processing fee.  The total cost to Jeff is:

STEP 1:  Jeff’s investment –> \$200

STEP 2:  Markup charged by the player –> .15 x \$200 = \$30

STEP 3:  Add totals from STEPS 1 & 2 –> \$200 + \$30 = \$230

STEP 4:  YouStake processing fee –> 8.3% x \$230 = 19.09

STEP 5:  Add totals from STEPS 3 & \$ –> \$230 + 19.09 = \$249.09.  This is the total amount Jeff would pay to YouStake

It gets expensive quickly!

Jeff understood that only \$200 can be used as a numerator, as the markup and processing fee don’t have anything to do with calculating his winnings.  Using a larger numerator of \$230 or \$249.09 would inflate the outcome.

The clarity he was seeking involved the denominator.  Should it be the full tournament buy-in of \$5,000?  Or the 40% of that being offered to investors (\$5,000 x 40% = \$2,000)?

Jeff called me because his player had revised the deal to reduce the portion of his tournament buy-in being offered to backers from 40% to 30%.  Should he now use a denominator of (\$5,000 x 30% = \$1,500)?  When the player changed the amount he was offering to backers, did that change’s Jeff’s share of ownership?

(No.)

This happens in poker math too.

Suppose you are calculating your equity in a poker hand.  You have K Q and the board shows A T♣ 8♠ 7.  You assume the villain has at least a pair of aces, so hitting a K or Q on the river won’t help.  You count 12 outs… 9 diamonds that can make a nut flush and 3 additional J’s that make a Broadway straight.  What is your equity?

In the best case, the numerator is 12.  You have that many outs.

Is the denominator 52, the total number of cards in the deck?  Or 50, after subtracting your cards?  Or 46, after further subtracting the community cards?  Or 44, after further subtracting the villain’s cards?  Or 30, after further subtracting two cards each for seven other players who started the hand?

If you can’t get the right answer (hint: your equity is 26.1%), check the denominator!

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