;; Rerum Cognoscere Causas II : In Which Divers Probabilities are Deduced Simply by Counting ;; The problem: Which e'dition's rules were used to generate a village? (def ^:dynamic *randomizer* (java.util.Random. 0)) (defn rand-int [n] (.nextInt *randomizer* n)) (defn D6 [] (inc (rand-int 6))) (defn three-D6 [] (reduce + (for [i (range 3)] (D6)))) (defn three-from-four-D6 [] (reduce + (drop 1 (sort (for [i (range 4)] (D6)))))) (defn mixed [] (if (zero? (rand-int 10)) (three-from-four-D6) (three-D6))) (defn first-edition [] {:str (three-D6) :int (three-D6)}) (defn second-edition [] {:str (mixed) :int (mixed)}) (defn third-edition [] (if (zero? (rand-int 10)) {:str (three-from-four-D6) :int (three-from-four-D6)} {:str (three-D6) :int (three-D6)})) (def village (binding [*randomizer* (java.util.Random. 0)] (doall (repeatedly 100 (case (rand-int 3) 0 first-edition 1 second-edition 2 third-edition))))) village ;-> ({:str 13, :int 18} {:str 11, :int 18} {:str 14, :int 15} {:str 6, :int 12} {:str 14, :int 13} {:str 18, :int 10} {:str 15, :int 11} {:str 12, :int 15} {:str 7, :int 8} {:str 16, :int 12} {:str 8, :int 7} {:str 9, :int 14} {:str 10, :int 9} {:str 11, :int 10} {:str 5, :int 10} {:str 7, :int 9} {:str 9, :int 13} {:str 12, :int 9} {:str 13, :int 9} {:str 5, :int 9} {:str 8, :int 13} {:str 9, :int 11} {:str 13, :int 14} {:str 12, :int 14} {:str 12, :int 17} {:str 14, :int 9} {:str 10, :int 11} {:str 18, :int 17} {:str 11, :int 9} {:str 8, :int 9} {:str 15, :int 13} {:str 8, :int 5} {:str 11, :int 9} {:str 10, :int 8} {:str 9, :int 12} {:str 5, :int 11} {:str 10, :int 7} {:str 9, :int 14} {:str 11, :int 9} {:str 11, :int 12} {:str 12, :int 13} {:str 15, :int 9} {:str 12, :int 12} {:str 6, :int 13} {:str 5, :int 4} {:str 12, :int 13} {:str 15, :int 10} {:str 14, :int 14} {:str 11, :int 4} {:str 12, :int 9} {:str 10, :int 12} {:str 7, :int 12} {:str 8, :int 11} {:str 10, :int 10} {:str 9, :int 8} {:str 8, :int 12} {:str 7, :int 9} {:str 13, :int 3} {:str 14, :int 9} {:str 8, :int 9} {:str 10, :int 11} {:str 15, :int 4} {:str 10, :int 11} {:str 8, :int 10} {:str 15, :int 10} {:str 8, :int 13} {:str 12, :int 5} {:str 8, :int 16} {:str 4, :int 8} {:str 10, :int 18} {:str 12, :int 12} {:str 11, :int 10} {:str 12, :int 8} {:str 12, :int 13} {:str 8, :int 12} {:str 9, :int 12} {:str 12, :int 10} {:str 15, :int 10} {:str 8, :int 11} {:str 7, :int 11} {:str 4, :int 8} {:str 12, :int 11} {:str 13, :int 9} {:str 14, :int 13} {:str 5, :int 9} {:str 17, :int 10} {:str 8, :int 13} {:str 9, :int 10} {:str 5, :int 14} {:str 15, :int 12} {:str 13, :int 13} {:str 11, :int 8} {:str 8, :int 6} {:str 12, :int 8} {:str 10, :int 3} {:str 14, :int 9} {:str 15, :int 12} {:str 15, :int 14} {:str 6, :int 10} {:str 16, :int 13}) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; The first thing that our philosophers should look at is the distribution of scores according to their models. ;; What do the results of rolling 3D6 and adding look like? We'll just ;; enumerate the 216 possible ways the dice can fall and count how ;; many ways there are to score the various possible numbers: (def threed6f (frequencies (for [i (range 1 7) j (range 1 7) k (range 1 7)] (reduce + [i j k])))) ;; {3 1, 4 3, 5 6, 6 10, 7 15, 8 21, 9 25, 10 27, 11 27, 12 25, 13 21, 14 15, 15 10, 16 6, 17 3, 18 1} ;; What about 4D6 and discarding the lowest? (def fourd6drop1f (frequencies (for [i (range 1 7) j (range 1 7) k (range 1 7) l (range 1 7)] (reduce + (drop 1 (sort [i j k l])))))) ;; {3 1, 4 4, 5 10, 6 21, 7 38, 8 62, 9 91, 10 122, 11 148, 12 167, 13 172, 14 160, 15 131, 16 94, 17 54, 18 21} ;; So the probability distributions for the dice rolling methods are: (defn p3d6 [char] (/ (threed6f char) (reduce + (vals threed6f)))) (defn p4d6drop1 [char] (/ (fourd6drop1f char) (reduce + (vals fourd6drop1f)))) (defn pmixed [char] (+ (* 9/10 (p3d6 char)) (* 1/10 (p4d6drop1 char)))) ;; Sanity checks! ;; As every schoolboy once knew: ;; You get 3 sixes only one time in ~200, so 18s are very rare (p3d6 18) ;-> 1/216 (p3d6 17) ;-> 1/72 ;; In fact only one roll in ten is in the 15-18 range: (reduce + (map p3d6 '(18 17 16 15))) ;; But you see 10 or 11 very often, in fact about a quarter of the time. (p3d6 10) ;-> 1/8 (p3d6 11) ;-> 1/8 ;; However if you get to roll four dice and ignore one of them, ;; then the high scores are more likely. (p4d6drop1 18) ;-> 7/432 (p4d6drop1 17) ;-> 1/24 (reduce + (map p4d6drop1 '(18 17 16 15))) ;-> 25/108 ;; And in fact you have a quarter chance of an exceptional score 15-18. ;; 10 and 11 are slightly less likely than they were, and 11 is more likely than 10 (p4d6drop1 11) ;-> 37/324 (p4d6drop1 10) ;-> 61/648 ;; Truly bad scores are very hard to get (p4d6drop1 3) ;-> 1/1296 ;; In fact only one in six scores are 'below average'. (reduce + (map p4d6drop1 '(3 4 5 6 7 8 9))) ;-> 227/1296 ;; However in the mixed distribution where you might be doing it one ;; way and you might be doing it the other, these differences are not ;; nearly as large: (pmixed 18) ;-> 5/864 (pmixed 3) ;-> 11/2592 ; A subtle bias indeed. (/ (pmixed 18) (pmixed 3)) ;-> 15/11 ;; Just as a sanity check, check that our three distributions add up to 1. (reduce + (map p4d6drop1 (range 3 19))) ;-> 1N (reduce + (map p3d6 (range 3 19))) ;-> 1N (reduce + (map pmixed (range 3 19))) ;-> 1N ;; Our three schools, with their different beliefs about how characteristics are generated, can now write down the chances ;; of combinations of characteristics: ;; The traditionalists of the first e'dition say that your chance of ;; getting strength 18, intelligence 18 is simply the chance of ;; getting both scores independently. (defn ptrad [{:keys [str int]}] (* (p3d6 int) (p3d6 str))) ;; Being truly gifted is terribly unlikely: (ptrad {:str 18 :int 18}) ;-> 1/46656 ;; As is being truly disadvantaged: (ptrad {:str 3 :int 3}) ;-> 1/46656 ;; But then, only 1 person in 16 is truly average: (reduce + (map ptrad (for [s '(10 11) i '(10 11)] {:str s :int i}))) ;-> 1/16 ;; Being a weedy genius is just as likely as being a genius who can bend swords. (ptrad {:str 3 :int 18}) ;-> 1/46656 ;; The second e'dition guys still model the two scores as independent things, (defn pindep [{:keys [str int]}] (* (pmixed int) (pmixed str))) ;; But they predict more beefy einsteins (pindep {:str 18 :int 18}) ;-> 25/746496 ;; And fewer weedy vegetables (pindep {:str 3 :int 3}) ;-> 121/6718464 ;; One good, one bad (pindep {:str 3 :int 18}) ;-> 55/2239488 ;; Has about the same frequency as in the traditionalists' model (apply / ((juxt pindep ptrad) {:str 3 :int 18})) ;-> 55/48 ;; The third e'dition guys think that you use either the traditional system for both characteristics, ;; or you use the enhanced system for both. They say that there is a common cause, being a 'player character'. ;; And they say that one in ten people are like that. (defn pcommon [{:keys [str int]}] (+ (* 9/10 (p3d6 int) (p3d6 str)) (* 1/10 (p4d6drop1 int) (p4d6drop1 str)))) ;; If we just look at the frequencies of a characteristic in ;; isolation, then we can't tell the difference between the second and ;; third schools. ;; They make the same guesses about the number of very strong people: ;; The chance of having STR 18 is the chance of having ;; STR 18 and any intelligence (reduce + (map pcommon (for [i (range 3 19)] {:str 18 :int i}))) ;-> 5/864 (reduce + (map pindep (for [i (range 3 19)] {:str 18 :int i}))) ;-> 5/864 (reduce + (map ptrad (for [i (range 3 19)] {:str 18 :int i}))) ;-> 1/216 ;; It's only by looking at both scores together that we can see ;; differences between what the world will look like if there's a ;; common cause and what it will look like if there isn't: ;; Both later e'ditions predict more supermen, but the common causers predict even more than the independents: (map #(int (* 1000000 (float %))) ((juxt pcommon ptrad pindep) {:str 18 :int 18})) ;-> (45 21 33) ;; And they both predict slightly fewer basket cases, but the common causers expect less of a drop. (map #(int (* 1000000 (float %))) ((juxt pcommon ptrad pindep) {:str 3 :int 3})) ;-> (19 21 18) ;; The traditionalists predict slightly more cases of people who are good at only one thing (map #(int (* 1000000 (float %))) ((juxt pcommon ptrad pindep) {:str 18 :int 3})) ;-> (20 21 24) (map #(int (* 1000000 (float %))) ((juxt pcommon ptrad pindep) {:str 3 :int 18})) ;-> (20 21 24) ;; Again, we ought to sanity check our distributions: (reduce + (map pcommon (for [i (range 3 19) j (range 3 19)] {:int i :str j}))) ;-> 1N (reduce + (map ptrad (for [i (range 3 19) j (range 3 19)] {:int i :str j}))) ;-> 1N (reduce + (map pindep (for [i (range 3 19) j (range 3 19)] {:int i :str j}))) ;-> 1N ;; Premature optimization is the root of all evil, and there is no ;; place to optimize more premature than an introductory tutorial ;; essay, but I am worried about my transistors wearing out if I have ;; to use these functions a lot. (def pcommon (memoize pcommon)) (def pmixed (memoize pmixed)) (def ptrad (memoize ptrad))

## Tuesday, March 5, 2013

### Rerum Cognoscere Causas II : In Which Divers Probabilities are Deduced Simply by Counting

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I think you would enjoy the probability distribution monad http://onclojure.com/2009/04/24/a-monad-tutorial-for-clojure-programmers-part-4/

ReplyDeleteThank you! I did.

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