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The Theoretical Probability of Hitting Space Mines Abstract Introduction Take your chances 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 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))
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)) =
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)) =
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
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
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