# 20

Document Sample

```					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)

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 9 posted: 11/20/2012 language: English pages: 6