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					20. The Three-Cornered Duel

  A, B and C are to fight a three-cornered pistol duel. All know that A's chance
of hitting his target is 0.3, C's is 0.5, and B never misses. They are to fire at their
choice of target in succession in the order A, B, C cyclically (but a hit man loses
further turns and is no longer shot at) until only one man is left unhit. What
should A's strategy be?

  We want to compute the probability that A does not get hit.
1. If A misses B or C, B must shoot at C, and then A must hit B.
    P(A does not get hit given that A misses first shot) = (1)(.3) = .3000
2. If A hits C, then B shoots at A and hits him.
    P(A does not get hit given that A hits C on first shot) = 0
3. If A hits B, then C and A alternately shoot at each other until one hits the other.
    We can simulate this as shown below. 1=Hit, 0=Miss

  Count      Shooter     Hit/Miss              Winner              Data Table             A did not get hit
    1          C             1                   C                         C                     11 times, or
    2                                                               2      C                 0.1618 of the time.
    3                                                               3      A
    4                                                               4      C              Theoretical prob of A not getting hit is 3/13
    5                                                               5      C                 0.2308
    6                                                               6      A
    7                                                               7      A
    8                                                               8      C
    9                                                               9      C
   10                                                              10      C
   11                                                              11      C
   12                                                              12      C
   13                                                              13      C
   14                                                              14      C
   15                                                              15      C
   16                                                              16      C
   17                                                              17      C
   18                                                              18      A
   19                                                              19      A
   20                                                              20      C
                                                                   21      C
                                                                   22      C
                                                                   23      C
                                                                   24      C
                                                                   25      A
                                                                   26      C
                                                                   27      C
                                                                   28      C
                                                                   29      A
                                                                   30      C
                                                                   31      A
                                                                   32      C
                                                                   33      C
                                                                   34      C
                                                                   35      C
36   C
37   C
38   C
39   C
40   C
41   C
42   A
43   C
44   C
45   A
46   C
47   C
48   C
49   C
50   C
51   C
52   C
53   C
54   C
55   C
56   C
57   C
58   A
59   C
60   C
61   C
62   C
63   C
64   C
65   C
66   C
67   C
68   C
  of the time.

al prob of A not getting hit is 3/13 =
20. The Three-Cornered Duel

  A, B and C are to fight a three-cornered pistol duel. All know that A's chance
of hitting his target is 0.3, C's is 0.5, and B never misses. They are to fire at their
choice of target in succession in the order A, B, C cyclically (but a hit man loses
further turns and is no longer shot at) until only one man is left unhit. What
should A's strategy be?


Summary:
 P(A does not get hit if he misses his first target) =           0.3000
 P(A does not get hit if he hits his first target (B)) =         0.2308


What should A do?
 He should try to miss his first target.
 Can he shoot into the ground or into the air?
20. The Three-Cornered Duel

  A, B and C are to fight a three-cornered pistol duel. All know that A's chance
of hitting his target is 0.3, C's is 0.5, and B never misses. They are to fire at their
choice of target in succession in the order A, B, C cyclically (but a hit man loses
further turns and is no longer shot at) until only one man is left unhit. What
should A's strategy be?

                                                                               
  The probability of A not getting hit if he hits B on his first shot is
                                                                               (. 5)[ (. 7 )(. 5)]
                                                                              k 0
                                                                                                      k
                                                                                                          (. 3)

  Here are the "partial sums" of the infinite series.                k        Sum
                                                                     0      0.15000
                                                                     1      0.20250
                                                                     2      0.22088
                                                                     3      0.22731
                                                                     4      0.22956
                                                                     5      0.23035
                                                                     6      0.23062
                                                                     7      0.23072
                                                                     8      0.23075
                                                                     9      0.23076
                                                                    10      0.23077
                                                                    11      0.23077
                                                                    12      0.23077
                                                                    13      0.23077
                                                                    14      0.23077
                                                                    15      0.23077
                                                                 infinity    3/13 =       0.230769
5 )] k (. 3)

				
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posted:11/20/2012
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