|
The Theoretical Probability of Hitting Space Mines
by Alejandro M. Dobniewski
Abstract
In war knowledge is power. A commander cannot make decisions without awareness
of the possible consequences and evaluation of his chances of success can make
the difference. On the basis of this information he will sometimes choose to
proceed and other times to retire, but to decide requires information. This
article is intended to try to aid understanding of mine hit probabilities by
which I mean the chances of hitting a mine while travelling through a minefield
in a ship.
Introduction
Calculating the probability of hitting a mine will aid you in making decisions
on whether to cross a mine field or not. In this article I plan to explain how
to handle the different types of probabilities and how those probabilities
differ for standard and non-standard games configurations.
Take your chances
It's not hard to find prebuilt tables of mine hit probabilities but they are
often poorly documented and confusing to anyone without a basic understanding of
probability. Those who do have some basic understanding aren't taught how to
calculate probabilities over distance or for anything other than the limited
configurations given. The truth is, however, that you don't need a Master's
Degree in Mathematics to understand probabilities or to figure how to calculate
them.
Maybe you are wondering what the problem is when trying to understand the
chance of something happening ... in such examples we must carefully define what
that 'something' is.
The Game
The host configuration has three parameters that can affect minehit probability:
- The probability of hitting one normal mine per light year travelled,
- The probability of hitting one web mine per light year travelled and;
- The probability of hitting one mine per light year travelled in a cloaked
ship.
In all cases the formulas are the same and the only change is the probability
of being hit per light year used. If you don't have the configuration, ask your
host about them ... personally, I wouldn't join a game without first reading the
configuration.
The Probability of Hitting a Mine Over A Given Distance
So how can we calculate the probability of hitting one mine when travelling a
given distance (x)? To do this we use a probability concept known as: the
binomial distribution. To put it simply, if you have some basic notion of
probabilities, it is obvious that chance of a minehit follow a binomial
distribution.
The probability of hitting a number x of mines when travelling d lightyears
is B(x,d,p) if p is the minehit probability per light year travelled.
The function B is therefore:
B(x,d,p) = (d/x) * (p^x) * ((1 - p)^(d-1))
Where (d/x) = d!/(x!*(d - x)!)
and n!=1*2*...*n
If you don't like this math don't worry, I don't either :-)
You don't really need to understand it, only how to use it. This is a
theoretical formula based on how minehits are supposed to work in host. Errors
in the host program could make this untrue however The Undead Hedgehog confirmed
this by experiment ... and we all know that hosts don't have bugs ;-)
The probability of minehit when you travel a light year is now known. It's
one of the configuration parameters a, b or c mentioned above. If the ship is
travelling in a WEB minefield you use c. If it's cloaked in a normal minefield
use b and in all other cases use a (d is the distance travelled within the
minefield). There is no use in calculating these figures outside the minefield,
you never will hit a mine there. For example when travelling 81 light years (20
through a minefield) then d=20.
Example:
Assume a 30 light year mine field, with normal configuration p=1% (which
means 0.01). We want to know what the probability is of crossing it without
minehits:
B(0,30,0.01) = (30/0) * (0.01^0) * ((1 - 0.01)^(30 - 0)) =
= (30! / (0!)*(30 - 0)!) * 1 * ( 0.99 ^ 30 ) =
= (30! / 1 * 30!) * 1 * (0.7397) =
= 1 * 1 * 0.7397 = 0.7397
Remember that for any number n^0= 1 and 0!=1.
But if the ship is cloaked:
B(0,30,0.01) = (30/0) * (0.005^0) * ((1 - 0.005)^(30 - 0)) =
= (30! / (0!)*(30 - 0)!) * 1 * ( 0.995 ^ 30 ) =
= (30! / 1 * 30!) * 1 * (0.8603) =
= 1 * 1 * 0.8603 = 0.8603
This result shows that you can use the shortcut formula (1-p)^d to calculate
the probability of NOT hitting a mine. Accordingly you might assume that the
chance of hitting 3 mines whilst travelling the same distance would be
calculated with the formula B(3,30,0.01). However B() is cumulative, in other
words, B(3,30,0.01) is the probability of hitting 0 mines, plus the probability
of hitting 1 mine plus the probability of hitting 2 mines plus the probability
of hitting 3 mines ... thus B(3,30,0.01) is the probability of hitting up to 3
mines. For exactly 3 hits you must reset B(2,30,0.01) to B(3,30,0.01) to remove
uncertainty. B(3,30,0.01), therefore, is the probability of hitting 0 or 1 or 2
or 3 mines OR the chance of hitting up to 3 mines.
The probability of hitting AT LEAST ONE mine is 1-B(0,30,0.01). Why? If
something has a 20% chance of happening then, logically, it has an 80% chance of
NOT happening. So, if the chance of not hitting a mine is B(0,30,0.01) then the
chance of hitting 1 or more mine is 100%-B(0,30,0.01). If you need to know the
probability of hitting more than 2 mines use 1- B(1,30,0.01).
PHost Configuration
PHost has more parameters that may affect the chance of hitting a mine:
| Parameter: |
MineOddsWarpBonusX100 |
| Effect: |
changes the probability of hitting a mine depending of the speed. The
higher the speed the higher the probability of a minehit. The bonus is reset
to each warp speed from 9 downwards, so, if p=1 and MineOddsWarpBonusX100 is
5 then: |
| Warp |
p |
| 9 |
1% |
| 8 |
0.95% |
| 7 |
0.9% |
| 6 |
0.85% |
| 5 |
0.8% |
| 4 |
0.75% |
| 3 |
0.7% |
| 2 |
0.65% |
| 1 |
0.6% |
| Parameter: |
MineTravelSafeWarp |
| Effect: |
Sets the speed under which p=0. If we take this to be 5 then the above
table would change as follows: |
| Warp |
p |
| 9 |
1% |
| 8 |
0.95% |
| 7 |
0.9% |
| 6 |
0.85% |
| 5 |
0 |
| 4 |
0 |
| 3 |
0 |
| 2 |
0 |
| 1 |
0 |
| Parameter: |
WebMineOddsWarpBonusX100
WebMineTravelSafeWarp |
| Effect: |
As for normal mines but for Webs |
So you can see that the only change is on the probability per light year (p).
Once we know p the results will be the same as in the previous section.
Summary
We have seen the formulas and how to apply them. It should be easy to use
them to check the chances of safely crossing a minefield. In the appendices I
have supplied a table for standard games to save you calculating the odds of a
minehit (it does assume that the game configuration is standard). Please note
that this data does NOT cover overlapping minefields ... possibly, someday, I'll
write another article to cover that but for now it is beyond the scope of this
article and I don't have time :-). Whether I do or don't I hope this article is
of some use to you.
Appendices
A table of the probabilities of hitting at least one mine
and hitting zero mines in a game of VGA Planets
(THost/Standard Configuration).
| |
Hitting 1 or more |
Not Hitting |
| Distance |
Normal
Ship |
Cloaked
Ship |
Normal
Ship |
Cloaked
Ship |
| 1 |
1,00% |
0,50% |
99,00% |
99,50% |
| 2 |
1,99% |
1,00% |
98,01% |
99,00% |
| 3 |
2,97% |
1,49% |
97,03% |
98,51% |
| 4 |
3,94% |
1,99% |
96,06% |
98,01% |
| 5 |
4,90% |
2,48% |
95,10% |
97,52% |
| 6 |
5,85% |
2,96% |
94,15% |
97,04% |
| 7 |
6,79% |
3,45% |
93,21% |
96,55% |
| 8 |
7,73% |
3,93% |
92,27% |
96,07% |
| 9 |
8,65% |
4,41% |
91,35% |
95,59% |
| 10 |
9,56% |
4,89% |
90,44% |
95,11% |
| 11 |
10,47% |
5,36% |
89,53% |
94,64% |
| 12 |
11,36% |
5,84% |
88,64% |
94,16% |
| 13 |
12,25% |
6,31% |
87,75% |
93,69% |
| 14 |
13,13% |
6,78% |
86,87% |
93,22% |
| 15 |
13,99% |
7,24% |
86,01% |
92,76% |
| 16 |
14,85% |
7,71% |
85,15% |
92,29% |
| 17 |
15,71% |
8,17% |
84,29% |
91,83% |
| 18 |
16,55% |
8,63% |
83,45% |
91,37% |
| 19 |
17,38% |
9,08% |
82,62% |
90,92% |
| 20 |
18,21% |
9,54% |
81,79% |
90,46% |
| 21 |
19,03% |
9,99% |
80,97% |
90,01% |
| 22 |
19,84% |
10,44% |
80,16% |
89,56% |
| 23 |
20,64% |
10,89% |
79,36% |
89,11% |
| 24 |
21,43% |
11,33% |
78,57% |
88,67% |
| 25 |
22,22% |
11,78% |
77,78% |
88,22% |
| 26 |
23,00% |
12,22% |
77,00% |
87,78% |
| 27 |
23,77% |
12,66% |
76,23% |
87,34% |
| 28 |
24,53% |
13,09% |
75,47% |
86,91% |
| 29 |
25,28% |
13,53% |
74,72% |
86,47% |
| 30 |
26,03% |
13,96% |
73,97% |
86,04% |
| 31 |
26,77% |
14,39% |
73,23% |
85,61% |
| 32 |
27,50% |
14,82% |
72,50% |
85,18% |
| 33 |
28,23% |
15,25% |
71,77% |
84,75% |
| 34 |
28,94% |
15,67% |
71,06% |
84,33% |
| 35 |
29,66% |
16,09% |
70,34% |
83,91% |
| 36 |
30,36% |
16,51% |
69,64% |
83,49% |
| 37 |
31,06% |
16,93% |
68,94% |
83,07% |
| 38 |
31,74% |
17,34% |
68,26% |
82,66% |
| 39 |
32,43% |
17,76% |
67,57% |
82,24% |
| 40 |
33,10% |
18,17% |
66,90% |
81,83% |
| 41 |
33,77% |
18,58% |
66,23% |
81,42% |
| 42 |
34,43% |
18,98% |
65,57% |
81,02% |
| 43 |
35,09% |
19,39% |
64,91% |
80,61% |
| 44 |
35,74% |
19,79% |
64,26% |
80,21% |
| 45 |
36,38% |
20,19% |
63,62% |
79,81% |
| 46 |
37,02% |
20,59% |
62,98% |
79,41% |
| 47 |
37,65% |
20,99% |
62,35% |
79,01% |
| 48 |
38,27% |
21,38% |
61,73% |
78,62% |
| 49 |
38,89% |
21,78% |
61,11% |
78,22% |
| 50 |
39,50% |
22,17% |
60,50% |
77,83% |
| 51 |
40,10% |
22,56% |
59,90% |
77,44% |
| 52 |
40,70% |
22,95% |
59,30% |
77,05% |
| 53 |
41,30% |
23,33% |
58,70% |
76,67% |
| 54 |
41,88% |
23,71% |
58,12% |
76,29% |
| 55 |
42,46% |
24,10% |
57,54% |
75,90% |
| 56 |
43,04% |
24,47% |
56,96% |
75,53% |
| 57 |
43,61% |
24,85% |
56,39% |
75,15% |
| 58 |
44,17% |
25,23% |
55,83% |
74,77% |
| 59 |
44,73% |
25,60% |
55,27% |
74,40% |
| 60 |
45,28% |
25,97% |
54,72% |
74,03% |
| 61 |
45,83% |
26,34% |
54,17% |
73,66% |
| 62 |
46,37% |
26,71% |
53,63% |
73,29% |
| 63 |
46,91% |
27,08% |
53,09% |
72,92% |
| 64 |
47,44% |
27,44% |
52,56% |
72,56% |
| 65 |
47,97% |
27,81% |
52,03% |
72,19% |
| 66 |
48,49% |
28,17% |
51,51% |
71,83% |
| 67 |
49,00% |
28,53% |
51,00% |
71,47% |
| 68 |
49,51% |
28,88% |
50,49% |
71,12% |
| 69 |
50,02% |
29,24% |
49,98% |
70,76% |
| 70 |
50,52% |
29,59% |
49,48% |
70,41% |
| 71 |
51,01% |
29,95% |
48,99% |
70,05% |
| 72 |
51,50% |
30,30% |
48,50% |
69,70% |
| 73 |
51,99% |
30,64% |
48,01% |
69,36% |
| 74 |
52,47% |
30,99% |
47,53% |
69,01% |
| 75 |
52,94% |
31,34% |
47,06% |
68,66% |
| 76 |
53,41% |
31,68% |
46,59% |
68,32% |
| 77 |
53,88% |
32,02% |
46,12% |
67,98% |
| 78 |
54,34% |
32,36% |
45,66% |
67,64% |
| 79 |
54,80% |
32,70% |
45,20% |
67,30% |
| 80 |
55,25% |
33,04% |
44,75% |
66,96% |
| 81 |
55,70% |
33,37% |
44,30% |
66,63% |
This article was submitted by the Editor of the, now defunct, E-Zine
Planeteer Resurrection.
Other articles, fiction & humour from the Planeteer Resurrection have been
submitted to the "UK Atheist
& Science E-Zine"
|