Download 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…..
• 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” ?
BT10
Ace 
2
BT10
3
4
BT10
5
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 ?
BT10
Ace 
BT10
2
3
4
BT10
5
6
Would you continue to play ?
Changed Scenario….
BT10
BT10
BT10
Ace 
5
6
2
3
4
-----
• What about now ?
Changed Scenario…FAIR ?
BT10
2
BT10
BT10
3
4
5
6
What about now ?
Changed Scenario ?
BT10
BT10
2
3
4
BT10
5
Warm Up Game….FAIR ?
BT*
Ace 
2
BT*
3
4
BT*
5
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
1
2
3
4
5
6
2
3
4
5
6
36 pairs of outcomes
of the type (i, j)
1 <= i, j <= 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
7
12
16
19
21
2
8
13
17
20
32
3
9
14
18
28
33
4
10
15
25
29
34
5
11
23
26
30
35
• What is so special about this collection ?
6
22
24
27
31
36
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….
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Choice & Chance : Paul Levy
American Mathematical Society
Uspensky
Feller
End of Part I