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Alice in Diceland. BIKAS K SINHA Faculty [1979-2011] INDIAN STATISTICAL INSTITUTE KOLKATA & Ex-Member [2006-2009] National Statistical Commission GoI. Alice in Diceland…. - PowerPoint PPT Presentation
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Alice in Diceland
BIKAS K SINHA Faculty [1979-2011] INDIAN STATISTICAL INSTITUTE KOLKATA & Ex-Member [2006-2009] National Statistical Commission GoI
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]
Warm-up Game….. “Racing Post” : LA-based News Paper “To Switch or Not to Switch” ? BT10 BT10 BT10 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 ?
Warm Up Game…FAIR ?
BT10 BT10 BT10
Ace 2 5 3 6 4
Would you continue to play ?
Changed Scenario….
BT10 BT10 BT10
Ace 2 5 3 ----- 6 4
• What about now ?
Changed Scenario…FAIR ?
BT10 BT10 BT10
2 3 5 4 6
What about now ?
Changed Scenario ?
BT10 BT10 BT10
2 3 5 4
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 ?
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 !!!
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 ?
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
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!!!
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 ?
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
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 ?
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
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
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……
References….
• Choice & Chance : Paul Levy
• American Mathematical Society
• Uspensky
• Feller
• End of Part I
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