The Usefulness of Thought Experiments
For this Introduction to Advantage Play Article, we're not actually going to be discussing any particular Advantage Play, but rather, we're going to be discussing the usefulness of thought experiments and sharpening our minds to become sharper gamblers.
Thought experiments can take many forms, but regardless of the form taken, they are all useful in keeping us mentally limber. One thought experiment is coming up with new games, whether they be table or machine, to help us hone our math skills. One such experiment that I did was to invent a Keno game that I call, "Locking Keno," with a unique Bonus feature:
1.) The player will select anywhere from 6-10 numbers and any pays that do not involve hitting the last ball will pay according to the paytable for the game.
2.) If the player hits the last ball on a WINNING game, then the last ball will turn into a Lock.
3.) The player will then be asked to hit the, "Draw," button with the word replaced with, "Spin," and the result will be the total number of locks, will shall be anywhere from 1-3 Locks.
4.) If the player Locks one or two additional numbers, then the game will randomly select which numbers get Locked in addition to the last ball hit.
5.) The game will then auto-play five Free Games with the locked balls in place drawing twenty additional numbers, the player will be paid according to the paytable based on the total number of hits.
4 Numbers: Pays 1 FOR 1
5 Numbers: Pays 2 FOR 1
6 Numbers: PAYS 6 FOR 1
7 Numbers: PAYS 50 FOR 1
8 Numbers: PAYS 150 FOR 1
9 Numbers: PAYS 1000 FOR 1
10 Numbers: PAYS 2000 FOR 1
(0.1473189 * 1) + (0.05142769 * 2) + (0.01147939 * 6) + (0.00161114 * 50) + (0.00013542 * 150) + (0.00000612 * 1000) + (0.00000011 * 2000) = 0.42626062
Probability of Bonus
0.0463854 (Would be the same as Cleopatra Keno or Fireball Keno)
ONE LOCKED BALL:
nCr(9,3)*nCr(70,17)/nCr(79,20) = 0.24957554327427117
nCr(9,4)*nCr(70,16)/nCr(79,20) = 0.11785511765729473
nCr(9,5)*nCr(70,15)/nCr(79,20) = 0.03428512513666756
nCr(9,6)*nCr(70,14)/nCr(79,20) = 0.006122343774404921
nCr(9,7)*nCr(70,13)/nCr(79,20) = 0.0006444572394110442
nCr(9,8)*nCr(70,12)/nCr(79,20) = 0.00003611182807044644
nCr(9,9)*nCr(70,11)/nCr(79,20) = 0.00000041480754228242
((0.24957554327427117) + (0.11785511765729473*2) + (0.03428512513666756 * 6) + (0.006122343774404921 * 50) + (0.0006444572394110442 * 150) + (0.00003611182807044644 * 1000) + (0.00000041480754228242*2000)) * 5 = 5.65361873598
TWO LOCKED BALLS
nCr(8,2)*nCr(70,18)/nCr(78,20) = 0.32798895156603053
nCr(8,3)*nCr(70,17)/nCr(78,20) = 0.22278494823353023
nCr(8,4)*nCr(70,16)/nCr(78,20) = 0.08767000277708364
nCr(8,5)*nCr(70,15)/nCr(78,20) = 0.020403200646303104
nCr(8,6)*nCr(70,14)/nCr(78,20) = 0.0027325715151298797
nCr(8,7)*nCr(70,13)/nCr(78,20) = 0.00019175940457051785
nCr(8,8)*nCr(70,12)/nCr(78,20) = 0.0000053725695246050255
THREE LOCKED BALLS
nCr(7,1)*nCr(70,19)/nCr(77,20) = 0.301797456341157
nCr(7,2)*nCr(70,18)/nCr(77,20) = 0.3308164425278066
nCr(7,3)*nCr(70,17)/nCr(77,20) = 0.18725459011007925
nCr(7,4)*nCr(70,16)/nCr(77,20) = 0.05895051910872865
nCr(7,5)*nCr(70,15)/nCr(77,20) = 0.010289545153523545
nCr(7,6)*nCr(70,14)/nCr(77,20) = 0.0009187093887074595
nCr(7,7)*nCr(70,13)/nCr(77,20) = 0.00003223541714763015
BONUS SPIN PROBABILITIES AND FINAL EXPECTED RETURN of FREE GAMES
ONE EXTRA BALL: 24/50
TWO EXTRA BALLS: 24/50
THREE EXTRA BALLS: 2/50
FREE GAMES RETURN * PROBABILITY
11.2630661629344 * 0.0463854 = 0.5224418291941774
0.5224418291941774+0.42626062 = 0.9487024491941773 or 94.87%
Ultimately, that was actually a really simple game to come up with. In fact, I could easily come up with a Progressive version of the game and determine the advantage points for hitting 9/10 or 10/10 numbers in the event that I decided that both results should pay the Progressive on a Ten-Spot.
However, an Advantage Player could also use existing games to come up with alterations and determine how those alterations would affect the House Edge of the game. For instance, a player could come up with a Blackjack alteration in which a Bonus of x:1 is offered based upon the player's initial two cards. The player could then compare the Expected Return of that particular result to the base House Edge of the game in order to determine a new house edge.
In addition to WizardofOdds.com and WizardofVegas.com, another gambling website that has already completed these thought experiments for you is APHeat.net. On this site, Eliot Jacobson, PH.D. has exhaustively analyzed the effects of being able to hole-card, or otherwise A.P. various games and side bets. Fortunately, the results of these exhaustive analyses are freely available on this site, and I think that all potential Advantage Players would have a better understanding of the mathematics of gambling if they were to go to this site and try to fully understand some of the advantage plays presented therein.
That site also has a good few posts regarding the mathematical affects of promotions that have been Advantage Played in the past. If you have ever looked at a potentially lucrative promotion but did not even know where to begin with respect to analyzing it, then you would do well to go to that site and look at some of the promotion-related posts to understand some of the mathematical concepts that a player must be aware of in order to analyze the potential lucrativeness of a promotion.
There is no one resource in which all of the answers to all of the questions will ever be handed to you. However, there are countless resources out there that will aide a player in understanding concepts and, "How-Tos," such that a player can use these concepts and eventually begin analyzing things for himself or herself.
It all begins and ends with using thought experiments to become sharper and smarter. Here's another good one: When confronted with a difficult play on Video Poker, yes, use the resources available at WizardofOdds.com in order to determine the best E.V. play for that particular hand. However, remember that hand and, when you are all done gambling for the day, you can go back and determine the probabilities associated with different potential hold decisions to learn why the play offered by WoO was the mathematically superior play.
In both Advantage Play and life, if we're not learning and thinking, we're dying.