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Ɂɧɚɹ ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ m ɩɪɢ ɤɨɬɨɪɨɦ ɜɟɪɨɹɬɧɨɫɬɶ pm ɜɵɢɝɪɵɲɚ
ɩɪɟɜɵɫɢɬ ɧɚɣɞɢɬɟ ɬɨɱɧɨɟ ɡɧɚɱɟɧɢɟ p
ɉɨɫɥɟ ɷɬɨɝɨ ɨɩɪɟɞɟɥɢɬɟ
m
ɧɚɢɦɟɧɶɲɟɟɱɢɫɥɨɩɨɜɬɨɪɟɧɢɣ ɨɩɵɬɚ ɩɪɢ ɤɨɬɨɪɨɦɫɣɧɚɞɟɠɧɨ
ɫɬɶɸɦɨɠɧɨ ɛɵɥɨ ɛɵ ɭɬɜɟɪɠɞɚɬɶ ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɜɪɟɡɭɥɶɬɚɬɟ ɦɨɞɟ
ɥɢɪɨɜɚɧɢɹɫɬɚɬɢɫɬɢɱɟɫɤɚɹɜɟɪɨɹɬɧɨɫɬɶɞɟɣɫɬɜɢɬɟɥɶɧɨɩɪɟɜɵɲɚɟɬ
ǮȇȋȇȔȏȌ 3 ǩȤșȕȓȎȇȋȇȔȏȏȔȌȕȈȜȕȋȏȓȕȕȘȉȕȏșȣȕȘȔȕȉȔȢȌȔȇȉȢȑȏ ȘȇȓȕȘșȕȦșȌȒȣȔȕȊȕ
ȓȕȋȌȒȏȗȕȉȇȔȏȦȉȌȗȕȦșȔȕȘșȔȢȜȕȖȢșȕȉȘȖȕȓȕȠȣȥ0DSOH
Ⱦɥɹɷɬɨɝɨɨɡɧɚɤɨɦɢɦɫɹɫɩɪɢɦɟɪɚɦɢ ɜɵɩɨɥɧɟɧɢɹ ɬɟɯ ɤɨɦɚɧɞ ɢ ɮɭɧɤ
ɰɢɣ0DSOHɤɨɬɨɪɵɟɩɨɧɚɞɨɛɹɬɫɹɩɪɢɜɵɩɨɥɧɟɧɢɹɷɬɨɝɨɡɚɞɚɧɢɹ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɢɡ n ɫɥɭɱɚɣɧɵɯ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ ɰɟɥɵɯ
ɱɢɫɟɥ ɜ ɰɟɥɨɱɢɫɥɟɧɧɨɦ ɞɢɚɩɚɡɨɧɟ >k m@ ɝɟɧɟɪɢɪɭɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɜɫɬɪɨɟɧɧɨɣ
ɩɪɨɰɟɞɭɪɵ
ɥɨɢɡɞɢɚɩɚɡɨɧɚ
rand(k..m)ɋɝɟɧɟɪɢɪɭɟɦɤ ɩɪɢɦɟɪɭ ɨɞɧɨ ɫɥɭɱɚɣɧɨɟɰɟɥɨɟɱɢɫ
> @
r:=rand(1..6): r();
ɑɬɨɛɵ ɫɝɟɧɟɪɢɪɨɜɚɬɶ n ɫɥɭɱɚɣɧɵɯ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ ɰɟɥɵɯ
ɱɢɫɟɥɧɟɨɛɯɨɞɢɦɨnɪɚɡɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɢɦɟɧɢɬɶɨɛɪɚɳɟɧɢɟ
randɗɬɨɦɨɠɧɨɫɞɟɥɚɬɶɫɩɨɦɨɳɶɸɨɩɟɪɚɬɨɪɚɰɢɤɥɚfor:
ɞɭɪɟ
r() ɤɩɪɨɰɟ
r:=rand(1..6): n:=7:
for i from 1 to n
do r() od;
ɉɨɦɢɦɨɨɩɟɪɚɬɨɪɚforɢɦɟɟɬɫɹɟɳɟɨɞɢɧɨɩɟɪɚɬɨɪɰɢɤɥɚɞɟɣɫɬɜɢɟɤɨɬɨ
ɪɨɝɨɛɭɞɟɬ ɩɨɧɹɬɧɨɢɡ ɫɥɟɞɭɸɳɟɝɨɩɪɢɦɟɪɚ ɜɤɨɬɨɪɨɦ ɪɚɜɧɨɦɟɪɧɨɝɟɧɟɪɢɪɭ
ɸɬɫɹɰɟɥɵɟɱɢɫɥɚɢɡɞɢɚɩɚɡɨɧɚ>±@ɞɨɬɟɯɩɨɪɩɨɤɚɧɟɩɨɹɜɥɹɟɬɫɹɨɬɪɢɰɚ
ɬɟɥɶɧɨɟɱɢɫɥɨ±
r:=rand(-1..3): a:=r();
while a>=0
do a:=r(): od;
a a a a a a a a a a ±
Ⱦɟɣɫɬɜɢɟɨɩɟɪɚɬɨɪɚɭɫɥɨɜɢɹifɛɭɞɟɬɩɨɧɹɬɧɵɦɢɡɫɥɟɞɭɸɳɟɝɨɩɪɢɦɟɪɚ
ɜ ɤɨɬɨɪɨɦ ɪɚɜɧɨɦɟɪɧɨ ɝɟɧɟɪɢɪɭɸɬɫɹ ɰɟɥɵɯ ɱɢɫɟɥ ɜ ɞɢɚɩɚɡɨɧɟ >± @ ɩɪɢ
ɷɬɨɦ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɤ ɝɟɧɟɪɢɪɭɟɦɨɝɨ ɱɢɫɥɚ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɢ
signum
ɢɡɚɬɟɦɜɵɜɨɞɢɬɫɹɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟɫɨɨɛɳɟɧɢɟɨɛɷɬɨɦɡɧɚɤɟ
r:=rand(-2..2):
for i from 1 to 8 do
a:=r(): b:=signum(a):
if b>0 then print(i,`-ɨɟ ɱɢɫɥɨ ɩɨɥɨɠɢɬɟɥɶɧɨɟ`)
else
if b=0 then print(i,`-ɨɟ ɱɢɫɥɨ ɧɭɥɟɜɨɟ`)
else print(i,`-ɨɟ ɱɢɫɥɨ
ɨɬɪɢɰɚɬɟɥɶɧɨɟ`)
fi:
fi:
od:
ɨɟɱɢɫɥɨɩɨɥɨɠɢɬɟɥɶɧɨɟ
ɨɟɱɢɫɥɨɨɬɪɢɰɚɬɟɥɶɧɨɟ
ɨɟɱɢɫɥɨɩɨɥɨɠɢɬɟɥɶɧɨɟ
ɨɟɱɢɫɥɨɨɬɪɢɰɚɬɟɥɶɧɨɟ
ɨɟɱɢɫɥɨɧɭɥɟɜɨɟ
ɨɟɱɢɫɥɨɩɨɥɨɠɢɬɟɥɶɧɨɟ
ɨɟɱɢɫɥɨɩɨɥɨɠɢɬɟɥɶɧɨɟ
ɨɟɱɢɫɥɨɩɨɥɨɠɢɬɟɥɶɧɨɟ
ɉɪɨɫɦɨɬɪɢɬɟɬɟɩɟɪɶɟɳɟɪɚɡɥɢɫɬɢɧɝɩɪɨɰɟɞɭɪɵ deMereɜɫɜɟɬɟɭɠɟɢɦɟ
ɸɳɢɯɫɹɭɜɚɫ ɡɧɚɧɢɣ ɨɛ ɨɩɟɪɚɬɨɪɚɯɰɢɤɥɚɢɭɫɥɨɜɢɹȿɫɥɢɚɥɝɨɪɢɬɦɡɚɥɨɠɟɧ
ɧɵɣɜɷɬɨɣ ɩɪɨɰɟɞɭɪɟ ɜɚɦɩɨɧɹɬɟɧɧɚɩɢɲɢɬɟ ɩɪɨɰɟɞɭɪɭ
CoinTossɩɨɡɜɨɥɹ
ɸɳɭɸ ɦɨɞɟɥɢɪɨɜɚɬɶ ɨɩɵɬ ɛɪɨɫɚɧɢɹ ɦɨɧɟɬɵ ɋ ɩɨɦɨɳɶɸ ɷɬɨɣ ɩɪɨɰɟɞɭɪɵ
ɧɚɣɞɢɬɟɜɟɪɨɹɬɧɨɫɬɶɜɵɩɚɞɟɧɢɹ + ɝɚɪɚɧɬɢɪɭɹɫɣ ɧɚɞɟɠɧɨɫɬɶɸɬɨɱɧɨɫɬɶ
ɧɚɣɞɟɧɧɨɝɨɪɟɡɭɥɶɬɚɬɚ
ǮȇȋȇȔȏȌ 4 ǩȤșȕȓȎȇȋȇȔȏȏȖȗȕȉȌȗȏȓȑȇȑȘȕȊȒȇȘȚȌșȘȦȕȖȗȌȋȌȒȌȔȏȌȊȌȕȓȌșȗȏȞȌȘȑȕȐ
ȉȌȗȕȦșȔȕȘșȏȘȗȌȎȚȒȣșȇșȇȓȏȓȕȋȌȒȏȗȕȉȇȔȏȦ
Ɋɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɢɣ ɨɩɵɬȾɚɧɚ ɤɪɭɝɨɜɚɹ ɤɚɧɚɜɤɚɜɞɨɥɶ ɨɤɪɭɠɧɨ
ɫɬɢ ɟɞɢɧɢɱɧɨɣ ɞɥɢɧɵ ɫ ɥɭɡɨɣ Ʌ ɂɡ ɥɭɡɵ Ʌ ɧɚɭɞɚɱɭ ɜɵɥɟɬɚɟɬ ɲɚɪɢɤ ɂɫɯɨɞ
ɨɩɵɬɚɪɚɫɫɬɨɹɧɢɟɨɬɥɭɡɵɅɧɚɤɨɬɨɪɨɦɨɤɚɠɟɬɫɹɲɚɪɢɤɩɨɫɥɟɜɵɥɟɬɚ
ȼɞɚɧɧɨɦ ɫɥɭɱɚɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɢɫɯɨɞɨɜ : >@ȼɫɟɢɫɯɨɞɵɫɱɢɬɚɸɬɫɹ
ɪɚɜɧɨɜɨɡɦɨɠɧɵɦɢ Ɍɪɟɛɭɟɬɫɹ ɜɵɱɢɫɥɢɬɶ ɱɚɫɬɨɬɭ ɩɨɩɚɞɚɧɢɹ ɲɚɪɢɤɚ ɧɚ ɞɭɝɭ
ɞɥɢɧɵ ȿ Ⱦɭɝɚ ɧɚ ɨɤɪɭɠɧɨɫɬɢ ɜɵɛɢɪɚɟɬɫɹ ɫɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ ɮɢɤɫɢɪɭɟɬɫɹ
ɬɨɥɶɤɨɟɟɧɚɱɚɥɨɜɬɨɱɤɟɅ
ɗɬɨɬɨɩɵɬ ɦɨɠɧɨ ɩɪɨɦɨɞɟɥɢɪɨɜɚɬɶɩɪɨɰɟɞɭɪɨɣ
ɦɟɬɪ Q ɡɚɞɚɟɬ ɱɢɫɥɨ ɩɨɜɬɨɪɟɧɢɣ ɨɩɵɬɚ
ɉɪɨɰɟɞɭɪɚ ProbBallɜɧɚɱɚɥɟɡɚɞɚɟɬ
ProbBall(n)ɝɞɟ ɩɚɪɚ
ɫɥɭɱɚɣɧɵɦɨɛɪɚɡɨɦ ɞɭɝɭɧɚ ɨɤɪɭɠɧɨɫɬɢ ɟɞɢɧɢɱɧɨɣ ɞɥɢɧɵ ɚ ɡɚɬɟɦ ɜɵɱɢɫɥɹɟɬ
ɱɚɫɬɨɬɭɩɨɩɚɞɚɧɢɹɧɚɷɬɭɞɭɝɭɜɵɥɟɬɚɸɳɟɝɨ©ɧɚɭɞɚɱɭªɢɡɥɭɡɵɲɚɪɢɤɚ
ɊɚɫɫɦɨɬɪɢɦɬɟɩɟɪɶɞɪɭɝɨɣɨɩɵɬȾɚɧɤɪɭɝɦɢɲɟɧɶɫɪɚɞɢɭɫɨɦɇɚɧɟɝɨ
ɧɚɭɞɚɱɭɛɪɨɫɚɸɬɞɪɨɬɢɤɉɨɩɚɞɚɧɢɹɜɬɨɱɤɢɤɪɭɝɚɫɱɢɬɚɸɬɫɹɪɚɜɧɨɜɨɡɦɨɠɧɵ
ɦɢ Ɍɪɟɛɭɟɬɫɹ ɜɵɱɢɫɥɢɬɶ ɱɚɫɬɨɬɭ ɩɨɩɚɞɚɧɢɹ ɞɪɨɬɢɤɚ ɜ ɧɟɤɨɬɨɪɭɸ ɫɥɭɱɚɣɧɨ
ɜɵɛɪɚɧɧɭɸɨɛɥɚɫɬɶɧɚɤɪɭɝɟ
ɗɬɨɬɨɩɵɬɦɨɠɧɨɩɪɨɦɨɞɟɥɢɪɨɜɚɬɶɩɪɨɰɟɞɭɪɨɣ
ProbDar(n,view)ɝɞɟQ±
ɱɢɫɥɨ ɩɨɜɬɨɪɟɧɢɣ ɨɩɵɬɚ ɗɬɚ ɩɪɨɰɟɞɭɪɚ ɫɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ ɧɚ ɦɢɲɟɧɢ ɮɨɪ
ɦɢɪɭɟɬɨɛɥɚɫɬɶ ɥɢɛɨɜ ɜɢɞɟ ɤɪɭɝɚ ɫ ɰɟɧɬɪɨɦ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ ɟɫɥɢɩɚɪɚɦ
ɟɬɪYLHZ ɥɢɛɨ ɜ ɜɢɞɟ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚɫ ɰɟɧɬɪɨɦ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬɟɫɥɢ
ɩɚɪɚɦɟɬɪYLHZ
Ɉɬɤɪɨɣɬɟɮɚɣɥ ȼɟɪɒɚɪɢɤȾɪɨɬɢɤPZV ɢɜɚɦɫɬɚɧɭɬɞɨɫɬɭɩɧɵɩɪɨɰɟ
ɞɭɪɵ ProbBallɢProbDar. & ɢɯ ɩɨɦɨɳɶɸ ɩɪɨɜɟɪɶɬɟ ɫɨɝɥɚɫɨɜɚɧ
ɧɨɫɬɶɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨɨɩɪɟɞɟɥɟɧɢɹɜɟɪɨɹɬɧɨɫɬɢɫɨɫɬɚɬɢɫɬɢɱɟɫɤɢɦ
ɈɬɤɪɨɣɬɟɮɚɣɥȽɢɩɨɬɟɧɭɡɚPZVɧɚɪɚɛɨɱɟɦɥɢɫɬɟɤɨɦɩɶɸɬɟɪɧɨɝɨɩɚ
ɤɟɬɚ0DSOHɜɵɧɚɣɞɟɬɟɩɪɨɰɟɞɭɪɭ
Hypo:=proc(n)
local r,a,b,c,i, total, test:
r:=uniform[0,1]: total:=0:
for i from 1 to n do
test:=true:
while test do
a:=r(): b:=r():
c:=sqrt(a^2+b^2):
if c<=1 then test:=false fi:
od:
if c<=0.5 then total:=total+1 fi:
od:
evalf(total/n):
end:
ȼ ɷɬɨɣ ɩɪɨɰɟɞɭɪɟ ɫɥɭɱɚɣɧɨ ɢ ɧɟɡɚɜɢɫɢɦɨ ɜɵɛɢɪɚɸɬɫɹ ɱɢɫɥɚ aɢbɢɡ
ɢɧɬɟɪɜɚɥɚ ɤɨɬɨɪɵɟ ɡɚɬɟɦ ɛɟɪɭɬɫɹ ɜ ɤɚɱɟɫɬɜɟ ɤɚɬɟɬɨɜ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ
ɬɪɟɭɝɨɥɶɧɢɤɚ Ɉɩɪɟɞɟɥɹɟɬɫɹ ɱɚɫɬɨɬɚ ɩɨɹɜɥɟɧɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɫ ɞɥɢɧɨɣ ɝɢɩɨ
ɬɟɧɭɡɵɧɟɩɪɟɜɨɫɯɨɞɹɳɟɣɈɩɵɬɩɪɨɞɨɥɠɚɟɬɫɹnɪɚɡɩɪɢɷɬɨɦɬɪɟɭɝɨɥɶɧɢ
ɤɢɭɤɨɬɨɪɵɯɝɢɩɨɬɟɧɭɡɚɩɪɟɜɨɫɯɨɞɢɬɢɝɧɨɪɢɪɭɸɬɫɹɢɜɪɚɫɱɟɬɧɟɩɪɢɧɢɦɚ
ɸɬɫɹ
ɉɨɩɪɨɛɭɣɬɟɩɪɟɞɫɤɚɡɚɬɶɱɚɫɬɨɬɭɩɨɹɜɥɟɧɢɹ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɫ ɞɥɢɧɨɣɝɢɩɨ
ɬɟɧɭɡɵɧɟ ɩɪɟɜɨɫɯɨɞɹɳɟɣ ɟɫɥɢ ɧɟ ɢɝɧɨɪɢɪɨɜɚɬɶɬɪɟɭɝɨɥɶɧɢɤɢ ɭ ɤɨɬɨɪɵɯ
ɝɢɩɨɬɟɧɭɡɚ ɩɪɟɜɨɫɯɨɞɢɬ ɉɪɨɜɟɪɶɬɟ ɫɜɨɣ ɩɪɨɝɧɨɡ ɫ ɩɨɦɨɳɶɸ ɩɪɨɰɟɞɭɪɵ
Hypoɩɪɟɞɜɚɪɢɬɟɥɶɧɨɦɨɞɢɮɢɰɢɪɨɜɚɜɟɟ
ǮȇȋȇȔȏȌ 5ǴȌȋȕȘșȇșȑȕȓȊȌȕȓȌșȗȏȞȌȘȑȕȊȕȕȖȗȌȋȌȒȌȔȏȦȉȌȗȕȦșȔȕȘșȏȎȇȞȇȘșȚȥȦȉȒȦȌș
ȘȦȔȌȉȕȎȓȕȍȔȕȘșȣȋȇșȣȕȈȡȌȑșȏȉȔȕȌȔȌȎȇȉȏȘȦȠȌȌȕșȘȖȕȘȕȈȇȗȇȘȞȌșȇȕȖȗȌȋȌȒȌȔȏȌ
ȉȌȗȕȦșȔȕȘșȏȘȕȈȢșȏȦǩȑȇȞȌȘșȉȌȖȗȏȓȌȗȇ ȗȇȘȘȓȕșȗȏȓ ȎȇȋȇȞȚǨȌȗșȗȇȔȇȑȕșȕȗȇȦȏȒ
ȒȥȘșȗȏȗȚȌș ȔȌȕȖȗȌȋȌȒȌȔȔȕȘșȣ ȖȕȔȦșȏȦ ªȔȇȚȋȇȞȚ« ȏȒȏ ªȘȒȚȞȇȐȔȕ« ȞȇȘșȕ ȉȘșȗȌȞȇȥ
ȠȚȥȘȦȉȊȌȕȓȌșȗȏȞȌȘȑȏȜȎȇȋȇȞȇȜ
ȼɤɪɭɝɟ ɪɚɞɢɭɫɨɦ ɧɚɭɞɚɱɭ ɫɥɭɱɚɣɧɨ ɜɵɛɢɪɚɟɬɫɹɯɨɪɞɚɄɚɤɨɜɚɜɟ
ɪɨɹɬɧɨɫɬɶɬɨɝɨɱɬɨɟɟɞɥɢɧɚɩɪɟɜɡɨɣɞɟɬɞɥɢɧɭɫɬɨɪɨɧɵɜɩɢɫɚɧɧɨɝɨɩɪɚɜɢɥɶɧɨ
ɝɨɬɪɟɭɝɨɥɶɧɢɤɚɬɟɩɪɟɜɡɨɣɞɟɬ
"Ɉɬɜɟɬɧɚɷɬɨɜɨɩɪɨɫɡɚɜɢɫɢɬɨɬɬɨɝɨɤɚ
ɤɨɣ ɫɦɵɫɥ ɜɤɥɚɞɵɜɚɟɬɫɹ ɜ ɫɥɨɜɨ ©ɫɥɭɱɚɣɧɨª ɱɬɨ ɜ ɫɜɨɸ ɨɱɟɪɟɞɶɡɚɜɢɫɢɬ
ɨɬɜɵɛɨɪɚɫɢɫɬɟɦɵɤɨɨɪɞɢɧɚɬ
ɉɪɨɫɬɪɚɧɫɬɜɨ ɢɫɯɨɞɨɜ ɡɞɟɫɶ ± ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɯɨɪɞ ɜ ɤɪɭɝɟ ȼɜɟɞɟɦ
ɩɪɹɦɨɭɝɨɥɶɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɫ ɧɚɱɚɥɨɦ ɜ ɰɟɧɬɪɟ ɤɪɭɝɚ ȼɫɩɨɦɧɢɦ ɱɬɨ
ɫɟɪɟɞɢɧɚ Ɇ ɧɟɤɨɬɨɪɨɣ ɯɨɪɞɵ Ⱥȼ ɥɟɠɢɬ ɧɚ ɪɚɞɢɭɫɟ ɟɣ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ
ɉɨɷɬɨɦɭ ɯɨɪɞɭ Ⱥȼ ɦɨɠɧɨ ɨɩɢɫɚɬɶ ɨɞɧɢɦ ɢɡ ɫɥɟɞɭɸɳɢɯ ɬɪɟɯ ɫɩɨɫɨɛɨɜ
ɩɪɹɦɨɭɝɨɥɶɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ x y ɬɨɱɤɢ Ɇ ɩɨɥɹɪɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚ
ɦɢrTɬɨɱɤɢɆ ɩɚɪɨɣ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬĮɢȕɤɨɧɰɨɜɯɨɪɞɵȺ
ɢȼȼɤɚɠɞɨɦɢɡɷɬɢɯɫɥɭɱɚɟɜɩɨɞɫɥɭɱɚɣɧɵɦɜɵɛɨɪɨɦɦɨɠɧɨɫɱɢɬɚɬɶɫɥɭɱɚɣ
ɧɵɣɜɵɛɨɪ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɨɨɪɞɢɧɚɬȼɵɱɢɫɥɢɦ ɢɫɤɨɦɭɸ ɜɟɪɨɹɬɧɨɫɬɶɦɨ
ɞɟɥɢɪɭɹɩɨɨɬɞɟɥɶɧɨɫɬɢɫɢɬɭɚɰɢɢ±ɫɥɟɞɭɸɳɢɦɨɛɪɚɡɨɦ
ɤɨɨɪɞɢɧɚɬɵxɢyɜɵɛɢɪɚɸɬɫɹɫɥɭɱɚɣɧɵɦ ɨɛɪɚɡɨɦ ɪɚɜɧɨɦɟɪɧɨ ɝɟɧɟɪɢ
ɪɭɸɬɫɹ ɧɚ ɨɬɪɟɡɤɟ > @ Ɂɚɬɟɦ ɩɪɨɜɟɪɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ x
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ɨɧɨɧɚɪɭɲɚɟɬɫɹ ɬɨ ɬɨɱɤɚ M xyɥɟɠɢɬ ɜɧɟ ɤɪɭɝɚɢ ɟɟ ɫɥɟɞɭɟɬɩɪɨɩɭɫɬɢɬɶ
ɜɩɪɨɬɢɜɧɨɦɫɥɭɱɚɟɬɨɱɤɚM xy±ɫɟɪɟɞɢɧɚɯɨɪɞɵɩɪɢɷɬɨɦɟɟɞɥɢɧɚɪɚɜɧɚ
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Ɂɚɦɟɬɢɦ ɱɬɨ ɥɸɛɨɣ ɩɨɜɨɪɨɬ ɤɪɭɝɚ ɜɨɤɪɭɝ ɰɟɧɬɪɚ ɧɟ ɢɡɦɟɧɹɟɬ ɞɥɢ
ɧɵ ɯɨɪɞɵ ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɩɪɟɞɩɨɥɚɝɚɬɶ ɱɬɨ ɯɨɪɞɚ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɩɚɪɚɥ
ɥɟɥɶɧɚɹɨɫɢɚɛɫɰɢɫɫɉɨɷɬɨɦɭ
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ɚrɜɵɛɢɪɚɸɬɫɹɫɥɭɱɚɣɧɵɦɨɛɪɚɡɨɦɪɚɜ
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ɧɨɦɟɪɧɨ ɝɟɧɟɪɢɪɭɸɬɫɹ ɧɚ ɨɬɪɟɡɤɟ >± @ Ⱦɥɢɧɚ ɯɨɪɞɵ ɩɪɢ ɷɬɨɦ ɪɚɜɧɚ
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ɉɪɟɞɩɨɥɨɠɢɦɱɬɨɨɞɢɧɤɨɧɟɰɯɨɪɞɵȺȼɫɤɚɠɟɦɬɨɱɤɚȼ ɥɟɠɢɬ ɜ ɬɨɱ
ɤɟɬɟȕ ɌɨɝɞɚĮɜɵɛɢɪɚɟɬɫɹɫɥɭɱɚɣɧɵɦɨɛɪɚɡɨɦɪɚɜɧɨɦɟɪɧɨɝɟɧɟ
ɪɢɪɭɸɬɫɹɧɚɢɧɬɟɪɜɚɥɟSɩɪɢɷɬɨɦɞɥɢɧɚɯɨɪɞɵ ɜ ɫɢɥɭ ɬɟɨɪɟɦɟ ɤɨɫɢɧɭ
ɫɨɜɪɚɜɧɚ
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Ɉɬɤɪɨɣɬɟɮɚɣɥ
ɉɚɪɚɞɨɤɫȻɟɪɬɪɚɧɚPZVɢɜɚɦɫɬɚɧɟɬɞɨɫɬɭɩɧɚɩɪɨɰɟɞɭɪɚ
BertranParadox(n,show ɦɨɞɟɥɢɪɭɸɳɚɹ ɜɫɟ ɬɪɢ ɜɚɪɢɚɧɬɚ ɜɵɛɨɪɚ ɫɢɫɬɟ
ɦɵ ɤɨɨɪɞɢɧɚɬ ɢ ɜɵɱɢɫɥɹɸɳɚɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɜɟɪɨɹɬɧɨɫɬɢ ɬɨɝɨ ɱɬɨ ɞɥɢ
ɧɚ ɧɚɭɝɚɞ ɜɵɛɪɚɧɧɨɣ ɯɨɪɞɵ ɩɪɟɜɡɨɣɞɟɬ
ɉɪɢ ɷɬɨɦ Q ± ɱɢɫɥɨ ɩɨɜɬɨɪɟɧɢɣ
ɨɩɵɬɚ ɚ ɩɚɪɚɦɟɬɪ VKRZ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ ɢɥɢ ɜ ɫɥɭɱɚɟ
VKRZ iɜɵɜɨɞɢɬɫɹ ɪɢɫɭɧɨɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ iɭɜɚɪɢɚɧɬɭɜɵɛɨɪɚɫɢɫɬɟɦɵɤɨ
ɨɪɞɢɧɚɬɜɫɥɭɱɚɟVKRZ ɪɢɫɭɧɤɢɧɟɜɵɜɨɞɹɬɫɹɜɨɜɫɟ
ɋ ɩɨɦɨɳɶɸ ɩɪɨɰɟɞɭɪɵ BertranParadox ɨɩɪɟɞɟɥɢɬɟ ɜɫɟɦɢ ɬɪɟɦɹ
ɫɩɨɫɨɛɚɦɢ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɞɥɢɧɚ ɧɚɭɝɚɞ ɜɵɛɪɚɧɧɨɣ ɯɨɪɞɵ
ɩɪɟɜɡɨɣɞɟɬ
ɉɪɢ ɷɬɨɦ ɡɚɞɚɣɬɟ ɬɚɤɨɟ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ n ɩɪɢ
ɤɨɬɨɪɨɦ ɫ ɣ ɧɚɞɟɠɧɨɫɬɶɸ ɦɨɠɧɨ ɝɚɪɚɧɬɢɪɨɜɚɬɶ ɱɬɨ ɪɟɡɭɥɶɬɚɬ
ɦɨɞɟɥɢɪɨɜɚɧɢɹɨɬɥɢɱɚɟɬɫɹɨɬɢɫɬɢɧɧɨɝɨɧɟɛɨɥɟɟɱɟɦɧɚ ɍɛɟɞɢ
ɬɟɫɶɱɬɨɜɫɟɬɪɢɜɟɪɨɹɬɧɨɫɬɢɛɭɞɭɬɪɚɡɥɢɱɧɵɦɢ
ɉɨɫɬɚɪɚɣɬɟɫɶɨɛɴɹɫɧɢɬɶɷɬɨɬɮɚɤɬ
Ɂɚɬɟɦɨɩɪɟɞɟɥɢɬɟ ɢɫɤɨɦɵɟ ɜɟɪɨɹɬɧɨɫɬɢɢɫɯɨɞɹɢɡɝɟɨɦɟɬɪɢɱɟɫɤɢɯɫɨ
ɨɛɪɚɠɟɧɢɣ
Ʉɚɤɨɣɠɟ ɢɡ ɬɪɟɯɫɩɨɫɨɛɨɜ ɛɨɥɶɲɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬɫɥɭɱɚɣɧɨɦɭ ɜɵɛɨɪɭ
ɯɨɪɞɵ"Ⱦɥɹ ɨɬɜɟɬɚɧɚ ɷɬɨɬɜɨɩɪɨɫ ɪɚɫɫɦɨɬɪɢɦ ɟɳɟ ɨɞɢɧ ɛɨɥɟɟ ɟɫɬɟɫɬɜɟɧɧɵɣ
ɫɩɨɫɨɛɜɵɛɨɪɚɫɥɭɱɚɣɧɨɣɯɨɪɞɵ
ɇɚ ɤɜɚɞɪɚɬɧɨɦ ɫɬɨɥɟ ɫ ɞɥɢɧɨɣ ɫɬɨɪɨɧɵ ɧɚɪɢɫɨɜɚɧ ɤɪɭɝ ɫ ɪɚɞɢɭɫɨɦ
ɰɟɧɬɪ ɤɨɬɨɪɨɝɨ ɫɨɜɩɚɞɚɟɬ ɫ ɰɟɧɬɪɨɦ ɫɬɨɥɚ ɢ ɹɜɥɹɟɬɫɹ ɰɟɧɬɪɨɦ ɩɪɹɦɨɭɝɨɥɶ
ɧɨɣɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɨɫɢ ɤɨɬɨɪɨɣ ɩɚɪɚɥɥɟɥɶɧɵɫɬɨɪɨɧɚɦ ɤɜɚɞɪɚɬɚ ɇɚ ɫɬɨ
ɥɟ ɫɥɭɱɚɣɧɨ ɪɚɜɧɨɦɟɪɧɨ ɜɵɛɢɪɚɟɬɫɹ ɬɨɱɤɚ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ x
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ɧɨ ɪɚɜɧɨɦɟɪɧɨ ɜɵɛɢɪɚɟɬɫɹ ɭɝɨɥ T ɢɡ ɢɧɬɟɪɜɚɥɟ
ɫɤɨɣɝɟɨɦɟɬɪɢɢɢɡɜɟɫɬɧɨɱɬɨɭɪɚɜɧɟɧɢɟɩɪɹɦɨɣx
ɬɨɦ k WJT ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɟɫɬɶ y y
ɩɪɹɦɨɣɞɨ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬɰɟɧɬɪɚɫɬɨɥɚɪɚɜɧɨ
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kx±x ɚ ɪɚɫɫɬɨɹɧɢɟ ɷɬɨɣ
ykx
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ɂɡ ɚɧɚɥɢɬɢɱɟ
ɉɨɷɬɨɦɭ ɹɫɧɨ
ɱɬɨ ɩɪɹɦɚɹ ɩɟɪɟɫɟɤɚɟɬ ɤɪɭɝ ɟɫɥɢ ɢ ɬɨɥɶɤɨ ɟɫɥɢ d ɉɟɪɟɫɟɤɚɹɫɶ ɫ ɤɪɭɝɨɦ
ɩɪɹɦɚɹɨɛɪɚɡɭɟɬ ©ɫɥɭɱɚɣɧɭɸª ɯɨɪɞɭɞɥɢɧɚ ɤɨɬɨɪɨɣ ɛɭɞɟɬɛɨɥɶɲɟ , ɟɫɥɢ ɢ
ɬɨɥɶɤɨɟɫɥɢ
d !
ɗɬɨɬɫɩɨɫɨɛɜɩɨɥɧɟɟɫɬɟɫɬɜɟɧɟɧɩɨɫɤɨɥɶɤɭɫɨɨɬɜɟɬɫɬɜɭɟɬɫɥɭɱɚɣɧɨɦɭɛɪɨ
ɫɚɧɢɸɫɨɥɨɦɢɧɤɢɧɟɨɝɪɚɧɢɱɟɧɧɨɣ ɞɥɢɧɵɧɚɧɟɨɝɪɚɧɢɱɟɧɧɭɸɩɥɨɫɤɭɸɩɨɜɟɪɯ
ɧɨɫɬɶɩɥɨɳɚɞɶɫɬɨɥɚɡɧɚɱɢɬɟɥɶɧɨɩɪɟɜɨɫɯɨɞɢɬɩɥɨɳɚɞɶɤɪɭɝɚɢɩɨɬɨɦɭɦɨɠɟɬ
ɫɱɢɬɚɬɶɫɹɧɟɨɝɪɚɧɢɱɟɧɧɨɣ
ɇɚɩɢɲɢɬɟ ɩɪɨɰɟɞɭɪɭ BertranNatural(n) ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɩɪɨɰɟɞɭ
ɪɨɣ Hypo) ɩɨɡɜɨɥɹɸɳɭɸ ɦɨɞɟɥɢɪɨɜɚɬɶ ɷɬɨɬ ɫɩɨɫɨɛ ɫɥɭɱɚɣɧɨɝɨ ɜɵ
ɛɨɪɚ ɯɨɪɞɵ ɢ ɫ ɟɟ ɩɨɦɨɳɶɸ ɧɚɣɞɢɬɟ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɞɥɢɧɚ
ɧɚɭɝɚɞ ɜɵɛɪɚɧɧɨɣ ɯɨɪɞɵ ɩɪɟɜɡɨɣɞɟɬ
ɉɪɢ ɷɬɨɦ ɡɚɞɚɣɬɟ ɬɚɤɨɟ
ɧɚɢɦɟɧɶɲɟɟ ɱɢɫɥɨ ɢɫɩɵɬɚɧɢɣ n ɩɪɢ ɤɨɬɨɪɨɦ ɫ ɣ ɧɚɞɟɠɧɨɫɬɶɸ
ɦɨɠɧɨ ɝɚɪɚɧɬɢɪɨɜɚɬɶ ɱɬɨ ɪɟɡɭɥɶɬɚɬ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɨɬɥɢɱɚɟɬɫɹ
ɨɬɢɫɬɢɧɧɨɝɨɧɟɛɨɥɟɟɱɟɦɧɚ
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DzȇȈȕȗȇșȕȗȔȇȦȗȇȈȕșȇ
ǶDzǵǹǴǵǸǹǯǴǬǶǷǬǷȂǩǴȂǼ
ǸDzǺǾǧǰǴȂǼǩǬDzǯǾǯǴ
ǯǺǸDzǵǩǴȂǬǩǬǷǵȆǹǴǵǸǹǯ
ǶȒȕșȔȕȘșȣȔȌȖȗȌȗȢȉȔȕȐȘȒȚȞȇȐȔȕȐȉȌȒȏȞȏȔȢ
ɉɭɫɬɶ ɏ ± ɧɟɩɪɟɪɵɜɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ȼɟɪɨɹɬɧɨɫɬɧɨɣ ɮɭɧɤɰɢɟɣ
PE ɡɚɞɚɧɧɨɣ ɧɚ ɦɧɨɠɟɫɬɜɟ ɫɨɛɵɬɢɣ ȿ ɧɟɩɪɟɪɵɜɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢɫɯɨ
ɞɨɜ : ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɹ PEɞɥɹɤɨɬɨɪɨɣɦɨɠɧɨɭɤɚɡɚɬɶɬɚɤɭɸɧɟɨɬɪɢ
f
xdx
ɰɚɬɟɥɶɧɭɸɮɭɧɤɰɢɸfxɨɬɜɟɱɚɸɳɭɸɭɫɥɨɜɢɸɧɨɪɦɢɪɨɜɤɢ
PE f xdx
³
E
ɫɹ ɩɥɨɬɧɨɫɬɶɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɟɩɪɟɪɵɜɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɏ ɟɫɥɢ
PX E PE f xdx
ɇɚɤɨɩɢɬɟɥɶɧɨɣ ɮɭɧɤɰɢɟɣ ɢɥɢ ɮɭɧɤɰɢɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɟɩɪɟɪɵɜɧɨɣ
ɫ ɜ ɏ ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɹ Fx ɨɩɪɟɞɟɥɹɟɦɚɹ ɬɚɤ Fx PXx ɞɥɹ ɥɸɛɨɝɨ
ɞɟɣɫɬɜɢɬɟɥɶɧɨɝɨɡɧɚɱɟɧɢɹɚɪɝɭɦɟɧɬɚx
ɋɨɨɬɧɨɲɟɧɢɹɦɟɠɞɭ ɧɚɤɨɩɢɬɟɥɶɧɨɣɮɭɧɤɰɢɟɣ Fx ɢ ɩɥɨɬɧɨɫɬɶɸ ɪɚɫɩɪɟ
ɞɟɥɟɧɢɹfxɧɟɩɪɟɪɵɜɧɨɣɫɜɏ
ɞɥɹɥɸɛɨɝɨɫɨɛɵɬɢɹE :ɉɪɢɷɬɨɦɮɭɧɤɰɢɹfxɧɚɡɵɜɚɟɬ
³
E
ɞɥɹɥɸɛɨɝɨɫɨɛɵɬɢɹE :
x
Fx ftdt
³
f
³
f
ɱɬɨ
Fcx fx
ǮȇȋȇȔȏȌ 1ǷȇȘȘȓȕșȗȏȓȘȒȌȋȚȥȠȏȐȕȖȢșǫȇȔȑȗȚȊȓȏȟȌȔȣȘȗȇȋȏȚȘȕȓr ǴȇȔȌ
ȊȕȔȇȚȋȇȞȚȈȗȕȘȇȥșȋȗȕșȏȑǶȕȖȇȋȇȔȏȦȉșȕȞȑȏȑȗȚȊȇȘȞȏșȇȥșȘȦȗȇȉȔȕȉȕȎȓȕȍȔȢȓȏ
ȄșȕșȕȖȢșȚȍȌȗȇȘȘȓȇșȗȏȉȇȒȘȦȉȒȇȈȕȗȇșȕȗȔȕȓȖȗȇȑșȏȑȚȓȌǩȞȇȘșȔȕȘșȏȈȢȒȕȚȘșȇ
ȔȕȉȒȌȔȕȞșȕ ȉȌȗȕȦșȔȕȘșȣȖȕȖȇȋȇȔȏȦȋȗȕșȏȑȇȉȖȗȕȏȎȉȕȒȣȔȚȥ ȕȈȒȇȘșȣEȗȇȉȔȇȕșȔȕ
S
ȟȌȔȏȥȖȒȕȠȇȋȏǬȑȖȒȕȠȇȋȏȉȘȌȐȓȏȟȌȔȏ
r S
ɉɪɟɞɩɨɥɨɠɢɦ ɬɟɩɟɪɶ ɱɬɨ ɧɚɫ ɢɧɬɟɪɟɫɭɟɬ ɬɨɱɧɨɫɬɶ ɛɪɨɫɤɚ ± ɪɚɫɫɬɨɹɧɢɟ
ɭɩɚɜɲɟɝɨ ɞɪɨɬɢɤɚ ɨɬ ɰɟɧɬɪɚ ɤɪɭɝɚ ɉɨɷɬɨɦɭ ɫɜɹɠɟɦ ɫ ɞɚɧɧɵɦ ɨɩɵɬɨɦ ɧɟɩ
ɪɟɪɵɜɧɭɸɫɥɭɱɚɣɧɭɸɜɟɥɢɱɢɧɭ ɏ ɜɵɪɚɠɚɸɳɭɸ ɪɚɫɫɬɨɹɧɢɟ ɭɩɚɜɲɟɝɨ ɞɪɨɬɢɤɚ
ɨɬ ɰɟɧɬɪɚ ɦɢɲɟɧɢ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɢɫɯɨɞɨɜ ɛɭɞɟɬ ɦɧɨɠɟɫɬɜɨ
aaS
: >@ɄɚɤɛɵɥɨɫɤɚɡɚɧɨɜɵɲɟPXa
PX b±PX a b
±a
S
Ɉɬɫɸɞɚ Pa X b
Ɍɟɩɟɪɶɫ ɩɨɦɨɳɶɸɤɨɦɩɶɸɬɟɪɧɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɧɚɣɞɟɦɩɥɨɬɧɨɫɬɶ ɪɚɫ
ɩɪɟɞɟɥɟɧɢɹɫɜ
ɝɪɚɦɦɭɫɜ
ɏȾɥɹɷɬɨɝɨ ɤɚɤ ɦɵ ɭɠɟ ɡɧɚɟɦ ɧɟɨɛɯɨɞɢɦɨ ɩɨɫɬɪɨɢɬɶ ɝɢɫɬɨ
ɏɤɨɬɨɪɚɹɢɛɭɞɟɬɩɪɢɛɥɢɡɢɬɟɥɶɧɨɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶɢɫɤɨɦɨɣɩɥɨɬ
ɧɨɫɬɢ
Ɉɬɤɪɨɣɬɟ ɮɚɣɥ
ɤɟɬɚ0DSOHɜɵɧɚɣɞɟɬɟɩɪɨɰɟɞɭɪɭ
ȾɪɨɬɢɤɉɥɨɬɧPZV ɧɚ ɪɚɛɨɱɟɦ ɥɢɫɬɟ ɤɨɦɩɶɸɬɟɪɧɨɝɨ ɩɚ
DartDensity
with(stats[random]):
read(`auxiliary proc for lab 2\\Dart_Density.m`):
with(Dart_Density);
>Gistogramma@
DartDensity:=proc(n,m)
local result,count,U,x,y,distance:
result:=[]: count:=1:
U:=uniform[-1,1]:
while count<(n+1) do
x:=U(): y:=U():
distance:=sqrt(x^2+y^2):
if distance<=1 then
result:=[op(result),distance]:
count:=count+1:
fi:
od:
Gistogramma(result,0,1,m);
end:
ɉɪɨɰɟɞɭɪɚ
DartDensity(n,k)ɫɬɪɨɢɬɝɢɫɬɨɝɪɚɦɦɭɧɚɨɬɪɟɡɤɟ>@N±
ɱɢɫɥɨɱɚɫɬɢɱɧɵɯɢɧɬɟɪɜɚɥɨɜ ɧɚ ɤɨɬɨɪɵɟ ɪɚɡɛɢɜɚɟɬɫɹɨɬɪɟɡɨɤ>@ Q ± ɱɢɫɥɨ
ɡɧɚɱɟɧɢɣɫɜ
ɏɝɟɧɟɪɢɪɭɟɦɵɯɧɚɩɪɨɦɟɠɭɬɤɟ>@
ɋɩɨɦɨɳɶɸɷɬɨɣ ɩɪɨɰɟɞɭɪɵ ɨɩɪɟɞɟɥɢɬɟɩɥɨɬɧɨɫɬɶ
ɫ ɜ ɏ Ɂɚɬɟɦ ɫ ɩɨɦɨɳɶɸ ɧɚɣɞɟɧɧɨɣ ɮɭɧɤɰɢɢ
Pa X b b
ǮȇȋȇȔȏȌ 2 ǷȇȘȘȓȕșȗȏȓ ȘȒȌȋȚȥȠȏȐ ȕȖȢșǫȇȔȇ ȑȗȚȊȕȉȇȦȑȇȔȇȉȑȇ ȉȋȕȒȣȕȑȗȚȍȔȕȘșȏ
ȌȋȏȔȏȞȔȕȐȋȒȏȔȢȘȒȚȎȕȐDzǯȎȒȚȎȢDzȔȇȚȋȇȞȚȉȢȒȌșȇȌșȟȇȗȏȑǵȘșȇȔȕȉȑȏȟȇȗȏȑȇȉ
șȕȞȑȇȜȑȇȔȇȉȑȏȘȞȏșȇȥșȘȦȗȇȉȔȕȉȕȎȓȕȍȔȢȓȏȄșȕșȕȖȢșȚȍȌȗȇȘȘȓȇșȗȏȉȇȒȘȦȉȒȇȈȕ
ȗȇșȕȗȔȕȓȖȗȇȑșȏȑȚȓȌ ǩȞȇȘșȔȕȘșȏ ȈȢȒȕ ȚȘșȇȔȕȉȒȌȔȕȞșȕ ȉȌȗȕȦșȔȕȘșȣȖȕȖȇȋȇȔȏȦ
ȟȇȗȏȑȇȔȇȖȗȕȏȎȉȕȒȣȔȚȥȋȚȊȚǬȗȇȉȔȇȕșȔȕȟȌȔȏȥȋȒȏȔȢȋȚȊȏǬȑȋȒȏȔȌȉȘȌȐȕȑȗȚȍ
ȔȕȘșȏȗȇȉȔȕȐ
±a
fxɪɚɫɩɪɟɞɟɥɟɧɢɹ
fxɜɵɜɟɞɢɬɟɮɨɪɦɭɥɭ
ɉɪɟɞɩɨɥɨɠɢɦɱɬɨ ɧɚɫ ɢɧɬɟɪɟɫɭɟɬɪɚɫɫɬɨɹɧɢɟ ɨɬ ɥɭɡɵ Ʌ ɧɚɤɨɬɨɪɨɦ ɨɤɚ
ɠɟɬɫɹɲɚɪɢɤ ɩɨɫɥɟɜɵɥɟɬɚɉɨɷɬɨɦɭɫɜɹɠɟɦɫ ɞɚɧɧɵɦɨɩɵɬɨɦɧɟɩɪɟɪɵɜɧɭɸɫ
ɏɜɵɪɚɠɚɸɳɭɸɪɚɫɫɬɨɹɧɢɟɨɫɬɚɧɨɜɢɜɲɟɝɨɫɹɲɚɪɢɤɚɨɬɥɭɡɵɅȼɷɬɨɦɫɥɭ
ɜ
ɱɚɟɩɪɨɫɬɪɚɧɫɬɜɨɦɢɫɯɨɞɨɜ ɛɭɞɟɬ ɦɧɨɠɟɫɬɜɨ
a
PX a
ɲɟ
a
ɈɬɫɸɞɚPa X b PX b±PX a b ±a
ɉɪɨɫɦɨɬɪɢɬɟ ɬɟɩɟɪɶ ɟɳɟ ɪɚɡ ɥɢɫɬɢɧɝ ɩɪɨɰɟɞɭɪɵ
ɢɦɟɟɬɫɹ ɨɩɟɪɚɬɨɪ
result:=[op(result),distance]: Ʉɨɦɚɧɞɚ
: >@Ʉɚɤɛɵɥɨɫɤɚɡɚɧɨɜɵ
DartDensity ȼ ɧɟɦ
op(llist)ɢɡɜɥɟɤɚɟɬɷɥɟɦɟɧɬɵɢɡɡɚɞɚɧɧɨɝɨɫɩɢɫɤɚlistɧɚɩɪɢɦɟɪ
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