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Alice in Diceland

Alice in Diceland. BIKAS K SINHA Faculty [1979-2011] INDIAN STATISTICAL INSTITUTE KOLKATA & Ex-Member [2006-2009] National Statistical Commission GoI. Alice in Diceland….

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Alice in Diceland

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  1. Alice in Diceland BIKAS K SINHA Faculty [1979-2011] INDIAN STATISTICAL INSTITUTE KOLKATA & Ex-Member [2006-2009] National Statistical Commission GoI

  2. 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]

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

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

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

  6. Changed Scenario…FAIR ? BT10BT10BT10 2 3 5 4 6 What about now ?

  7. Changed Scenario ? BT10BT10BT10 2 3 5 4

  8. Warm Up Game….FAIR ? BT*BT*BT* 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 ?

  9. 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 !!!

  10. 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 BT 100.00 from the Opponent. • Is it a FAIR Game ?

  11. Sample Space… • 1 2 3 4 5 6 • 1 • 2 • 3 36 pairs of outcomes • 4 of the type (i, j) • 5 1 <= i, j <= 6 • 6

  12. Choice & Chance !!! • Opponent : II III IV I • Self : I II III ? Computations : P[ I dominates II ] = 67 % P[ II dominates III ] = 67 % P[ III dominates IV ] = 67 % Conclusion : ‘I’ BEST & ‘IV’ Worst !!! Q. Winning Strategy ? Ooooppppsssss!!!

  13. 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 ?

  14. 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

  15. 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 ?

  16. 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

  17. 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

  18. 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……

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

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