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