Some Mathematics of Machine Gaming

1. Some Mathematics of Machine Gaming Prepared for American Mathematics Association of Two Year Colleges in Minneapolis for New Orleans 2 November 2007 By Robert N. Baker New Mexico State University-Grants rnbaker@nmsu.edu **This version for the AMATYC website has been edited to reduce file size**

2. My thesis: You don’t need to rig the machine to make profit, just rig the numbers. The latter is legal, the former is not. This workshop investigates how to do so, in classroom ready activities, with keystrokes for using table features of the TI -83 model graphing calculator.

3. The Deck: A Set to Model Computer Memory The Standard Deck of Cards and the Algebra of Manipulating Computer Memories at the MAA at Reed College, 1996 ** ** Dynamic Memories: S. Brent Morris, NSA Director Serial access and “the Perfect Shuffle” ** ** Static Memories: aka RAM me, grad student Direct access and set construction **Naming and accessing locations In an economical fashion **Constructing large blocks on demand From non-contiguous locations Originally the deck was designed as a physical model of the ancient four-tier Caste Social System of China, < 1000 AD Pips evolved to indicate things associated with the four classes in different societies across Eurasia, through about 1600 AD. Peasants (clubs) Military (spades) Professional (diamonds) Theological (hearts)

4. In 2007, decks of cards, on sale literally around the world, use the pips and colors and structure established by 1600 in France. Historical Note: The deck called the “tarot” spun off the standard deck sometime after 1400’ish, in Italy.

5. Expansion-enabling technology for deck < 1000 A.D. Chinese paper and printing - Block printing on tiles, heavy paper - Popular among royalty (invented by ? 969 A.D. ?) < 1300 World trade routes - Italy, Marco Polo, the Silk Road, the Turks - Muslim expansion, Iberian peninsula mines < 1400 Renaissance Art, rise of Leisure, intellectual discourse < 1500 Johann Gutenberg’s printing press (Cash for a starving artist!) < 1600 Christian expansion, Central American mines, more world trade routes < 1800 New Orleans, Mississippi River trade, riverboat culture of movement - Improved printing, cutting, stock - Rounded corners eased shuffling! - Factory system kept price down. 1800s America moved west in wagons - “Corner indicators” printed for shortcut - New! Blank card(s) in each deck free. < 1870 - Players adopted blank as wild card < 1871 - Joker (from tarot Jester?) introduced < 1875 Improved deck as model of real society: The one of every bunch who fits nowhere and everywhere.!?

6. Poker • A “vying” game; winner determined by players comparing their combinations of cards • Evolved in parlors and around campfires around the world for more than 500 years • Formalized in cosmopolitan 1800s New Orleans with a 20 card deck for 4 players max. • Now a television sport! (my San Diego topic) • Now also a one-player machine game--the topic of today’s workshop.

7. Poker across continents and centuries: A “combination” game ? 969 A.D. Chinese emperor Mu Tsung & wife (inventress?) < 15th Century Persian game As ras 15th Century Regional European variations similar in structure - Italian Il Frusso, Primiera; French LaPrime, L’Amigu, L’Mesle or Brelan - English Post and Pair and Brag; German Pochen < 1700 Cardano, Pascal, Fermat, Huygens establish theory of probability 1708 -P. R. de Montmort, Essai d’analyse sur les jeux de hazard (Analtyic Essay on Games of Chance). Applied probability to card games and all life. 1718 Louisiana territory game Poque - Five card hand from 20 card deck, one bet & showdown 1800 Faro and 3-card Monte still most popular card games on the circuit 1834 J. H. Green documented “the cheating game,” by the name Poker ≈ 1840 Full-deck Poker introduced - Lowered typical hand; enabled more players per game - The “flush” arose outside of royal setting, accepted 1860s 5-card “Draw” poker introduced--a second bet - 5-card “Stud” poker introduced--four bets possible - “Straight” became an officially-valued hand - “Bluffing” entered as strategy - Faro and 3-card Monte lost favor on the circuit 1870 “Jacks or Better” introduced, also known as “Jack-pots” - Added more bets, increased single pots, strategy ≤ 1875 The new Joker employed as a wild card “Low ball” and “split pot” games introduced - Increased pot sizes, number of winners 1930s “7-card stud” popularity rose above 5-card draw’s (Better: fewer cards “down” makes cheating harder, no discards to watch.) 1931 Nevada legalized games of chance

8. Defining Order on the set Comparative-value for the deck’s “kinds” In the rules of poker, the kinds are given a (transitive) ordering. HIGHToLOW Aces, kings, queens, jacks, 10’s, … , 3’s, 2’s A > k > q > j > 10 > 9 > … > 3 > 2 This post-1800 ordering is used to determine the winner of the cut, and to help well-define an order on the hands. (Some house rules allow the ace also low.)

9. Defining Order in the game Comparative-value for the types of hands In the rules of poker, the types of hands are given a (transitive) ordering. HIGHToLOW Straight Flush, 4 of a kind, full house, flush, straight, 3 of a kind, two pair, one pair, none of the above. SF > 4K > FH > Fl > St > 3k > 2pair > 1pair > nota This ordering is used to determine the winner of each round of play (also known as a hand), nested on the ordering of the kinds.

10. In poker, the value/ordering of hands is given by “the rules of the game.” Circa 15th Century It is not an arbitrary order. Two hands compare inversely to the probabilities of getting them (dealt from a randomized deck) Circa 18th Century Higher valued hands have lower probability of occurrence.

11. Essential thesis: Roll the Bones: The history of gambling by David G. Schwartz The elaboration of probability allowed for another path: using a discrepancy between the true odds and actual payouts to carve out a statistically guaranteed profit. This is the most significant change in all of gambling history and directly let to lotteries, bookmaking, and casinos. Thanks to a better understanding of probability, professional gamblers can now offer casual players the chance to bet as much as they liked against an impersonal vendor, with the “house odds” the irreducible price of entertainment. Pg. 80: 2006 Gotham Books, Penguin Group, Inc.

12. We will look at how to exploit this difference between odds and payout, beginning with: Probabilities of different poker hands Probabilities can be computed for five-card hands in straight poker from a randomized 52-card deck, by counting 5-card subsets of the deck and using techniques from probability theory.

13. General Definition Applied: The probability you get a certain type, call it type E, of valued combination in your 5 cards from a randomized deck, ie. hand, is defined in general by: Pr (E) = the number of different ways of satisfying E the number of different 5-card hands in the deck

14. First identify the denominator … the number of “different” 5-card hands. - ie. the number of 5-element subsets of a 52-element set - ie. the number of ways to choose 5 items from 52 By the multiplication rule for events, there are: 52 • 51 • 50 • 49 • 48 = 311,875,200 ways to be dealt five cards from the deck.

15. But this computation implies a concern for the process, which is not reflected by rules for the game of poker. Poker values only the finished five-card product (subset, combination), not the order in which they arrived (permutation). What adjustment is needed to compute the number we want?

16. Investigation: Let a poker hand include a 1, 2, 3, 4, and 5. List all of the ways this one poker hand can result, written “first-card-dealt on left, to last-card-dealt on right”. Then count the number of these different permutations of these five cards that all lead to the same poker hand (combination).

17. A method:Keep as much as constant as possible for as long as possible, to make exhaustive list.   12345 13245 14235 15234 12354 13254 14253 15243   12435 13425 14325 15324 12453 13452 14352 15342   12534 13524 14523 15423 12543 13542 14532 15432   _________________________________________ 2 • 3 + 2 • 3 + 2 • 3 + 2 • 3 ways = (2 • 3) • 4 = 4! different permutations listed so far, all with the 1 dealt first.

18. (Solution Cont.) This list includes only the ways to hold this hand where the ace was the first card received by the player. Four different but similar lists--with the 2 listed first, 3 first, the 4 and 5--yield a total of 5 lists for these five cards, each with 24 different orderings. Thus we have constructed the (2 • 3 • 4) • 5 = 5! = 120 ways to be dealt this hand.

19. With (2 • 3 • 4) • 5 = 5! = 120different ways to be dealt this one particular poker hand, we generalize to say the same is true for any poker hand. Thus, we can assert (311,875,200) / 120 = 2,598,960 different five-card hands are possible to construct from a standard deck of 52 cards.

20. Second, identify the numerators: The numbers of each of the types of handsfrom a standard deck of 52 playing cards. Each computation represents its own rationale for approaching the given counting problem. There are often several different ways to discover the one absolute answer to “how many ways can you ...?” In the following, “aCb” stands for the standard computation a!/[(a-b)! b!].

21. Numbers of Hands 1 pair, no better i) 13 • 4C2 •12C3 •4^3 1,098,240 ii) (52 • 3 / 2!) (48 • 44 • 40 / 3!)   2 pairs, no better i) 13C2 • 4C2 • 4C2 • 11 • 4 123,552 ii) [(52 • 3 / 2) (48 • 3 / 2) / 2!] • 44 iii) [(13C1 • 4C2) (12C1 • 4C2) / 2!] • 11C1 • 4C1 3 of a kind, no better i) 13 • 4C3 • 12C2 • 4^2 54,912 ii) (52 • 3 • 2 / 3!) (48 • 44 / 2!) iii) 13 • 4C3 • (12 • 4C1 • 11 • 4C1 / 2!) Straight, no better i) 10 • 4^5 - 40 10,200 (there are 40 “straight flush” hands!) Flush, no better i) 13C5 • 4 - 40 5,108 Full-house i) 13 • 4C3 • 12 • 4C2 3,744 ii) (52 • 3 • 2 / 3!) (48 • 3 /2!)  

22. Numbers of Hands, cont. 4 of a kindi) 13 • 4C4 • 12 • 4C1624 ii) (52 • 3 • 2 • 1 / 4!) • 48 Straight Flush i) 10 • 440 None of the Above, “nota”: 1,302,540 for these two methods, you need some of the above information i) (hands without even a pair) - (hands that are flush OR straight) [(13 • 4) (12 • 4) (11 • 4) (10 • 4) (9 • 4) / 5!] - (10,200 + 5,108 + 40) = 1,317,888 - 15,348 = ii) (total number of hands) - (hands noted above as valued) 2,598,960 - (40 + 624 + 3744 + 5108 + 10,200 + 54,912 + 123,552 + 1,098,240) = 2,598,960 - 1,296,420

23. Sum of numbers of different hands = 2,598,960 This number agrees with the value independently computed via 52 choose 5.

24. The Probabilities for 5-card poker hands from a randomized 52-card deck. The probability of an event =(# of elements in that event)___ (# of elements in the sample space) Then straight poker probabilities fall from the defining formula: The probability of a hand = (# of ways that hand can occur) 2,598,960 For ease of writing, let s = 2,598,960 the number of different 5-card hands.

25. Type of hand MethodProbability...and no better. Straight Flush 40/s ≈ .00001539 4 of a kind 624/s ≈ .00024010 Full house 3,744/s ≈ .00144058   Flush 5,108/s ≈ .00196540   Straight 10,200/s ≈ .00392464 3 of a kind 54,912/s ≈ .02112845   Two pairs 123,552/s ≈ .04753902 One pair 1,098,240/s ≈ .42256903   None of the above 1,302,540/s ≈ .50117739

26. Two tools needed in the move to machines: Definition: A random variable is a function with a non-numeric domain (often well-defined situations) and numeric range. It is a rule that assigns a number to a condition. Often defined with a function table, a random variable is neither random nor variable. Definition: The expected value of a random variable is a weighted average of the random variable’s values (range), with their probabilities as the weights. For random variable called $ with range values $i then ExpVal($) = ∑ $i • Pr ( $i )

27. Cooperative Data Entry: Observed on the helm of USCG M/V Planetree, 2001 To transfer data surely and quickly: Use two pairs to accommodate the human propensity for typos. On deck at the source of data: • One person reads the data out loud. • One person reads and listens, to catch spoken typos. In the helm at the destination of data: • One person types the data into the machine. • One person listens and watches, to catch written typos.

28. To a single-player game … • Winning determined by comparing the player to the true odds, rather than to other players. • Winnings determined from the true odds, rather than by vying. • In a “fair game” each hand paid at the reciprocal of its probability. (Our first activity.) • Commercial machine games are designed to generate profit at specified rates, typically capped by state laws. (Our second activity.)

29. A Problem: Logistical and psychological difficulties arise in paying out in Single-Player games. Our Solution: Define and adjust a random variable until it meets given constraints. I’ll call it “payout” and denote it “$” For this we will use the power of the list in TI-83 calculators

30. First, we’ll use the number 2,598,960 often, so want to give it an easy name. I like the letter “s.” In the home screen of your calculator (use 2nd QUIT to get there, from anywhere) type 2 5 9 8 9 6 0 STO then ALPHA S then ENTER(S is the letter on the LN button, in the leftmost column.) From here on, use 2nd RCL ALPHA s to invoke 2,598,960.

31. Next, we’ll begin work with lists. Hit STAT then ENTER to get into the lists window. We’ll want to use at least 6 lists, so if already full of data, you should clear them.

32. ** Indicates auxiliary calculator info, often relevant to students. ** To clear all lists, use 2nd MEM ClrAllLists. MEM is an option on the + key. Once in the MEM window, you may simply hit the 4 key to activate 4: ClrAllLists, or else you may arrow down to highlight 4: ClrAllLists and then hit ENTER. Either way, the calculator will go to its home screen, awaiting your command to execute that operation. Hit ENTER. Then return to the list screen via STAT ENTER.

33. ** Indicates auxiliary calculator info, often relevant to students. ** To clear one entire list, arrow up until the list title is highlighted, hit CLEAR then hit the down arrow. That list will be emptied, awaiting data input, without altering the contents of any other list.

34. ** Indicates auxiliary calculator info, often relevant to students. ** To remove one entry from a list, use arrows to highlight it, then hit DEL. This will remove that value, and move the rest of the list up by one position. ** To change one entry in a list, highlight it, key in your desired value, then hit ENTER.

35. In L1 enter the basic data for our investigation: The “number of ways” each type of hand can occur. Arrow the cursor into L1 then type: 40 ENTER 624 ENTER 3744 ENTER 5108, 10200, 54912, 123552, 1098240, 1302540. Be sure to hit ENTER after each value.

36. In L2, compute probabilities. Right arrow into L2, then up arrow until title is highlighted. Type 2nd L1 ÷ ALPHA S ENTER L2 fills with decimal approximations for the probabilities of each of the types of hands

37. ** Indicates auxiliary calculator info, often relevant to students. ** Recall: 1.5E–5 is the TI’s format for scientific notation, and means 0.000015. ** To view an entry with more accuracy, highlight that entry, look on bottom row of screen

38. Use L3 to compute a fair game payout rule: In theory, in a fair game the payout for an event should be inversely proportional to its probability. Thus we can create a reasonable random variable for payout by taking the reciprocals of probabilities. Right arrow into L3, then up arrow until title is highlighted. Type 1 ÷ 2nd L2 ENTER L3 fills with the reciprocals of the probabilities

39. In computing the expected value of this payout design, each product-pair is a number times its reciprocal (pr times 1/pr), thus 1. With a partition of 9 elements, the expected value of this random variable is 9. For our goal of a fair game, with our random variable of inverses, the cost to play should be 9.

40. This, however, is not convenient to consumers. We can adjust our random variable’s values to yield an expected value of 1, with commensurate cost to play of 1, by simply dividing each of its values by 9.  

41. Use L4 to construct a better random variable. Right arrow into L4, then up arrow until title is highlighted. Type 2nd L3 ÷ 9 ENTER . L4 fills with payout values for a conjectured fair game with cost to play of 1.

42. We can now use features of the calculator to find the expected value for the suggested payouts, ie. for the random variable $. To determine E ($) = ∑ ($ • P ($)) we need determine the products $ • P ($) and then sum them.

43. The traditional paper and pencil approach requires that we compute each of the nine products x times P(x), organize these in a table, and then add them to obtain the random variable’s expected value. This method can be duplicated with the lists in the calculator. Use the lists that already contain the given information: L2 has the probabilities, L4 has the suggested “fair game” payout assignments.

44. To duplicate the traditional method: In LIST EDIT screen, arrow into L5, then up arrow until title is highlighted. Type 2nd L2 * 2nd L4 ENTER . L5 fills with the desired products x * P (x). To obtain the sum of these products, and thus the expected value of the game, we then can: Use the STAT CALC features of the calculator.

45. Tradition continued: In Home Screen, compute E($) by summing the list of products in L5. Type 2nd QUIT to get to home screen, then STAT then arrow right to CALC then ENTER = 1 for the 1-Var Stats summary. (This places you back in home screen, where you are prompted to supply the location for a list of data.) Type 2nd L5 ENTER to obtain basic stats on data in L5. Look on the second line from the top of this summary, ∑ xgivesthe sum of the products stored in L5, which is the desired “expected value”.

46. OR to obtain that sum of products more directly: The calculator can also give us our desired result more directly—using its “sum” feature. From the home screen, hit: 2nd LIST right arrow to MATH then type 5. This places sum( on the home screen. This command needs a list for its argument. Type 2nd L5 ) Enter. The returned sum is the desired “expected value.” Is it the same as ∑ x found in the first method?

47. Yet another way to obtain the sumfrom the home screen, without needing to prepare L5 and its intermediate products. In the home screen, type 2nd ENTRY , edit the argument to “Sum (L2•L4)”. Hit Enter.   This should yield the same expected value as found above. It provides a method to obtain the sum of products, without documenting those products themselves.   Some keystrokes given later in this activity use this time-saving method, when trying via guess and check to discover a random variable that will address legal concerns as well as human psychology in the programming of gaming machines.

49. Problem: The payouts for a fair game with cost of 1 include decimal fractions. This is inconvenient; the machine and players want payouts only in whole number multiples of the cost to play. Note: No nice sample size smaller than 100,000 plays enables us even to expect a straight flush in the sample. In the above table, I chose to round probabilities to 5 decimal places to ease working with sample sizes 100,000.  

50. Activity 1: In search of expected value of 1 with whole number payout values. 1) In L5, round to whole number values to approximate the values in L4.   2) Check the expected value by the following key strokes, for sum of Probs • Payouts: 2nd QUIT 2nd LIST arrow right to MATH then hit 5 = Sum( 2nd L2 * 2nd L5 ) ENTER