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Alice in Diceland

Alice in Diceland. BIKAS K SINHA Retired Professor of Statistics INDIAN STATISTICAL INSTITUTE KOLKATA & Ex-Member [2006-2009] National Statistical Commission, GoI ***********

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Alice in Diceland

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  1. Alice in Diceland BIKAS K SINHA Retired Professor of Statistics INDIAN STATISTICAL INSTITUTE KOLKATA & Ex-Member [2006-2009] National Statistical Commission, GoI *********** RUDS Spl. Talk : Sept. 12, 2017

  2. Quotes of the Day…. A Man’s Feet Should Be Planted In His Home Country ....BUT His Eyes Should Survey The World ! • ********** • Ignorance shouts ! • • Knowledge speaks !! • • Wisdom listens !!! • ********** • • Discuss what you know Today….. • • Use what you know Today AND • Figure out the rest……Don’t wait for • when you think - you will be Ready……

  3. Dr. Marepalli Bhaskara Rao[PhD- ISI -1973:Advisor Ashoke Moitra] Inspired by his talk at 1st Iranian Int’l Stat Conf in 1991

  4. Alice in Diceland….. • Fun & Game Unbounded.. • As soon as Alice Landed – • In a Mysterious Diceland ! • Magician started The Show For all….HIGH & Low……. • ĎĬČĘ Ĝãɱëš……..All fun… • & So Much to Learn…. AND so many challenges… With the Games of Chance !!! [Bikas K. Sinha, ISI, Kolkata]

  5. Warm-up Game….. “Racing Post” : LA-based News Paper “To Switch or Not to Switch” ? Rs.10Rs.10Rs.10 Ace  3 5 2  4 6 FAIR 6-faced DICE Cash Reward against Cash Entry Fee ENTRY FEE : Rs . 10 /- IS THIS A FAIR GAME ? Gambling : Fair or Unfair ?

  6. Warm Up Game…FAIR ? Rs.10Rs.10Rs.10 Ace  2 5 3 6 4 Would you continue to play ?

  7. Changed Scenario…. Rs.10Rs.10Rs.10 Ace  2 5 3 ----- 6 4 • What about now ?

  8. Changed Scenario…FAIR ? Rs.10Rs.10Rs.10 2 3 5 4 6 What about this game ?

  9. Changed Scenario…. Rs.20Rs.10 Ace  2 3 4 5 6 What about this game ?

  10. Changed Scenario ? Rs.10Rs.10Rs.20 2 3 5 4 • What about this ?

  11. Changed Scenario…. Rs.20Rs.20 3 5 4 Would you like to continue ?

  12. Warm Up Game….FAIR ? Rs.*Rs.*Rs.* Ace  3 5 2  4 6 Possible Scenario : All the Money [Rs. 30] in exactly one box…..other two are empty ! Rs. 30 -- -- -- Rs. 30 -- -- -- Rs. 30 To Switch OR Not To Switch the Choice ?

  13. Dice Game I [Hungarian Brothers’ Puzzle] Four Hungarian Brothers Honest BUT Very Special !!! [Indian Adaptation : Names Changed !] • Bore Bhaia : 4 4 4 4 0 0 • Du-Numbari : 3 3 3 3 3 3 • Tisree Kasam : 2 2 2 2 6 6 • Chhote Golam: 5 5 5 1 1 1 Non-Transtitive Dominance !!!

  14. Dice Game I • No Entry Fee ! • You Choose “One Dice” & I do next. • We BOTH Throw our Chosen Dice to check WHO got a Larger Number on the Upper-most Face of the Dice….Winner must show a Larger Number and will receive Rs. 100.00 from the Opponent. • Is it a FAIR Game ?

  15. Sample Space… • F a c e s • 1 2 3 4 5 6 • F 1 • a 2 • c 3 36 pairs of outcomes • e 4 of the type (i, j) • s 5 1 <= i, j <= 6 • 6

  16. Details ….. • D \ Bore Bhaia • 0 4 4 4 4 0 0 • N 3 < < < < > > • U 3 < < < < > > • M 3 < < < < > > • B 3 < < < < > > • A 3 < < < < > > • R 3 < < < < > > • I BB beats DN …chance =24/36=67%

  17. Choice & Chance !!! • Opponent : DN TK CG BB • Self : BB DN TK ? Computations : P[ BB dominates DN ] = 67 % P[ DN dominates TK ] = 67 % P[ TK dominates CG ] = 67 % Conclusion : ‘BB’ BEST & ‘CG’ Worst !!! Q. Winning Strategy ? Ooooppppsssss!!!

  18. Dice Game II : Nagpur Version • Courtesy : Professor M N Deshpande • Institute of Science, Nagpur There are 6 dice.....with the following compositions : I II III IV V VI ***************************************************************************** • 1 2 3 4 5 6 • 7 8 9 10 11 22 • 12 13 14 15 23 24 • 16 17 18 25 26 27 • 19 20 28 29 30 31 21 32 33 34 35 36 • What is so special about this collection ?

  19. Go to ….. • Page 29

  20. Sample Space….. • Once more 36 pairs of outcomes when two dice are compared • Dice I • 1 7 12 16 19 21 • D 2 • I 8 • C 13 36 pairs of outcomes • E 17 • 20 • II 32

  21. Dice Game II : Dominance…. P [ II Dominates I ] • = P [ III Dominates II ] • = P [ IV Dominates III ] • = P [ V Dominates IV ] • = P [ VI Dominates V ] = 21 / 36 > 50 % • P[ VI Dominates I ] = 5/6 + 1/36 = 31/36 • Is it a Fair Game ?

  22. Card Games…. • Full Pack ….shuffled ….draw cards one by one…note the colors [Red / Black] and put back : sampling WITH REPLACEMENT • Betting on “NO TWO SUCCESSIVE OUTCOMES ARE RED” !!! # Draws : 2 3 4 5 6 Wining Chance : 3/4 5/8 8/16 13/32 21/64

  23. Probability Computations…. • Two Cards Randomly Drawn • Sample Space : Color Combinations (R, R) (R, B) (B, R) (B, B) Bold : Favourable ……Chance = ¾ Three Cards Randomly Drawn Sample Space…….8 color combinations (R,B,R) (R,B,B) (B,R,B) (B,B,B) (B,B,R) (R,R,R) (R,R,B) (B,R,R) : Bold Fav…5/8

  24. Card Games : Frobenius Numbers Sequence ....0, 1, 1, 2, 3, 5, 8, 13, 21, ….. F_0, F_1, F_2, F_3, ….. F_(n+1) = F_(n-1) + F_(n) F# = Sum of Last Two F #’s P[No Two Successively Red out of n Cards] = P_n = F_(n+2) / 2^n Same for Black Cards……

  25. References…. • Choice & Chance : Paul Levy • American Mathematical Society • Uspensky • Feller • End of Part I

  26. Part II Elementary Statistical Inference Go to Page 40

  27. How many bacteria in the jar ? • Capture – Recapture Technique : Innovative Statistical Method for ‘ascer-taining’ the size [N] of a finite population Demonstration with Marbles…..same size and shape…..almost same color…..no distinguishing features as such.... Q. How many are there ?

  28. Capture-Recapture [CR] Method • ‘Capture’ a few items (k) and ‘Mark’ them and ‘Release’ in the population. • 2. Next Recatch AT RANDOM a few (n) Items & Count the Number (X) ‘Recaptured’. • N = Population Size [unknown] • k = Initial Catch Size [for Marking] • n = Random Catch Size [Pre-Fixed]

  29. Capture-Recapture [CR] Method • X = No. ‘Marked” items in the chosen sample • Population Proportion of “Marked” = k/N • Sample Proportion of “Marked” = X / n • “Estimating Equation” : k / N = X / n • Implies : N^ = kn / X • Q. What if “X = 0 ‘” ? ….N^ = Infinity !!! • Compromise : N^ = k(n+a)/a with a >0.

  30. Estimation of N…. • k \ n • 10 20 • 5 X = 2, N^ = 25 X = 3, N^ = 34 • 10 X = 2, N^ = 50 X = 3, N^ = 67

  31. Ascertaining the Size of a Finite Population : CMR Method • 1. ‘Capture’ a few items (k) : ‘Mark’ & ‘Release’ in the population. • 2. Recatch one-by-one & Inspect & Release UNTIL Initially Marked Items are Recaptured ‘m’ times • N = Population Size [unknown] • k = Initial Catch Size [for Marking] • n = Second Catch Size UNTIL Marked Items are Recaptured ‘m’ times [m being prespecified] • N^ = kn/m

  32. Estimation of Size of a Finite Population…. • k \ m 2 3 5 n = 15 n = 25 N^ = 2.5n = 38 N^ = 1.67n = 42 2 3 10 n = 8 n = 13 N^ = 5n = 40 N^ = 3.3n= 44

  33. CMR Method : Modified…. • Recapture one-by-one & Inspect & Release …..BUT….Keep Aside the Marked Items ….STOP as soon as ‘m’ Marked Items are found in the process of sampling. • N^ = {(k+1)n/m} – 1

  34. Estimation of N…. • k \ m 2 3 5 n = 9 n = 17 N^ = 3n - 1 N^ = 2n - 1 = 26 = 33 10 n = 7 n = 13 N^ = 5.5n - 1 N^ = 3.67n - 1 = 38 = 47

  35. Size of a Finite Population… • Sequential Search…..CMRR Method • Capture One Item -Mark & Release • Recapture & Inspect : Stop [if Already Marked] Mark & Release [if NOT Already Marked] Continue until one Marked is Discovered s = No. of attempts made after First Entry N^ = s(s+1) / 2……..

  36. Size of a Finite Population… • s : 1 2 3 4 5 ……10 • N^ : 1 3 6 10 15 …….55 Q. What if more items [k] are marked initially ? N^ = (s+k+1)_c_2 – k_c_2 k = 5 s = 1 2 3 4 5……. N^ = 11 18 26 35 45……

  37. New Game….. • Estimation of Total Number of Units produced….in a production process…. • Units Serially Numbered as 1, 2, … • No Omission of numbers ….. • No Duplication of numbers….. • How far does it go ?

  38. Marbles : Serially Numbered ? Natural Numbering : 1, 2,…,N? How many ? Pick one marble : Holds the number ’21’ “Best” Judgment for N if we just stop here?

  39. Next Draw….. ’11’….. BAD NEWS ? N^ = ???

  40. And….next ’5’ …..Ooooopppsssss! N^ = ???

  41. Next….. ’26’ …………Great !!! N^ = ?

  42. And….next ….. ’9’……what’s this…..most erratic ! N^ = ???

  43. And….. ’30’…Better ‘stop’ Random Choice? N^ = ???

  44. ‘Best’ Guess for ‘N’ ? • Guess Value of N • 21….. ? • 21 ...11…. ? • 21…11…5… ? • 21..11..5.. 26… ? • 21…11…5…26…9… ? • 21…11..5…26…9….30…. ?

  45. Thought Process…. • Concept of Partitioning of Popl. Units.. • Median : 50 % cut-off value • Upper 50 % : ………..X..……….[1/2] • Upper 67 % : ………x……..X………[2/3] • Upper 75 % : ……x…..x…..X…..[3/4] • Upper 80 % : …..x…..x…..x…..X….[4/5] • Upper 83 % : …x…x…x…x…X…[5/6] • And so on…….[n/(n+1)] at the n-th stage • N. n/(n+1) = X = Max. Value….N^ =X(n+1)/n

  46. ‘Best’ Guess for ‘N’ …. Guessed Value of N • 21….. 42 • 21 …11…. 32 • 21…11…5… 28 • 19..11..5.. 26… 33 • 19…11…5…26…9… 31 • 19…11..5…26…9….30…. 35 Q. Why ‘waste [?]’ all other information… Q. Is there any ‘extra’ information in the rest, beyond what is captured by the largest number ? …..Decisively NOT…..except for how many are there …the sample size [n]…

  47. The End…. • That’s it from • BKS……. • RUDS • Special Talk • September 12, 2017

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