# The Secret Formula

A couple of week's agi, my dealt with explaining what a full-pay machine is.  A reader had stated that he didn't know what that meant and how to find one, so I explained it.  But taking one step even further back might be helpful to some of you.  To that end, I mean explaining what exactly is the meaning of 'payback' and how it is calculated.

First, let's begin with a definition of the word payback.  Payback is the percent of the total wager that the Player can expect to have returned to him when he plays a particular game.  The critical piece here is the total amount he wagers.  The amount you buy in for is meaningless.  We need to add up EVERY Dollar (or nickel or dime or quarter) that you put at risk at any point in a game for the entire time you play.  If you're playing video poker, and you're playing the same amount every hand, the calculation of your total wager is very easy.  It is the number of hands you played times the amount you wagered per hand.  So, if you played 700 hands of max-coin quarters, you wagered 700 times \$1.25 or \$875.

If you're playing blackjack, the calculation is not so simple.  You would have to keep track of how many times you Doubled Down and Split.  On average, you wager about 1.14 times the base wager when playing Blackjack.  So, you've been wagering \$10, you're really wagering about \$11.40 per hand.  If you were to play 50 hands, we could approximate your total wager at \$570.    If you were to play a game like Ultimate Texas Hold'em, you'd have to keep track of how many times you Played 4x, 2x, 1x and Folded.  You would include your Ante and Blind Wagers.  You would NOT include your Trips wager as this is an independent wager with its own payback.

Ultimate Texas Hold'em also has an average wager per hand associated with it.  I believe it is between 4 and 5 units. This means a \$5 Player is really about a \$22 per hand Player.  But this assumes using Expert Strategy.  Since we know that many Players are scared away from betting 4X as often as they should, you're average wager per hand may be less than this.  So, to get a more accurate picture it might be better to keep tabs on your own total wager.

Back to our video poker example.  A Player wagers \$875 and is playing a game with a 99.5% payback.  This means he will lose 0.5% of his total wager, or about \$4.38.  Having just played 700 hands, it is possible that his actual loss will be this, but this is just a theoretical amount that is more accurate over the long run.  The long run doesn't have to be years, but it's probably more than an hour or two.  The more hands you play, the more likely your actual results will match the expected ones.

Blackjack and Ultimate have similar paybacks to video poker.  But, over 50 hands in blackjack (probably about the same amount of time as it takes to play 700 hands of video poker), the \$10 Player will wager about \$570.  With a 0.5% house advantage (100% MINUS the payback), the Player can expect to lose \$2.85.  A \$5 Player at UTH playing 50 hands would wager something closer to \$1100.  This will translate to a \$5.50 loss or nearly twice as much as a Blackjack Player wagering twice as much as a 'base'.  But Blackjack has a 1.14 average unit wager as compared to UTH's 4+.  As a result, you'll lose more playing UTH, on average.

So, how is the payback actually calculated.  There are a variety of methods that are used to get us to where we need to be.  But, in the end, it comes down to multiplying the payout of each winning possibility by the frequency (probability) of that winning possibility and summing up these values.  Yeah, that's confusing and is a lot easier when I show an example.  Let's use one of the simplest possible scenarios.  You're playing Roulette and just put money on number 22!  What is the payback of this wager?  Assuming a Double 00 wheel, if 22 comes up, you'll win 36 to 1.

So, what is the probability of number 22 coming up?  There are 38 numbers on the wheel and each has an equal probability.  So 1 of 38 wins and 37 of 38 loses.  The probability of winning is 1/38 or 2.6316%.  The payout is the TOTAL amount returned to the Player if he wins.  36 to 1 means the Player will win \$36 for each \$1 wagered and keep is original wager.  Thus is payout is 36 PLUS 1 (the original wager) for a total payout of 37.

We know multiply the payout of the winning possibility by the frequency of that possibility.  This is 37 times 1/38 or 37/38, which is 97.3684%  That is the payback of wagering on a single number on Double 00 roulette wheel.  It's that simple.  Well, not every case is THAT simple.  Next week, I'll go through more examples, which will include more complex wagers.