\u8868\u304c\u51fa\u308b\u78ba\u7387 \\(a \\ \\left( 0 \\lt a \\lt \\dfrac{1}{2} \\right)\\) , \u88cf\u304c\u51fa\u308b\u78ba\u7387\u304c \\(1-a\\) \u306e\u30b3\u30a4\u30f3\u3092 \\(1\\) \u679a\u6295\u3052\u308b\u8a66\u884c\u3092 \\(n\\) \u56de\u884c\u3046.\r\n\u305f\u3060\u3057 \\(n \\geqq 2\\) \u3068\u3059\u308b. \u3053\u306e \\(n\\) \u56de\u306e\u8a66\u884c\u306e\u7d50\u679c, \u8868\u304c \\(2\\) \u56de\u4ee5\u4e0a\u51fa\u308b\u4e8b\u8c61\u3092 \\(A _ n\\) \u3067\u8868\u3059. \u307e\u305f \\(1\\) \u56de\u76ee\u304b\u3089 \\(n\\) \u56de\u76ee\u306e\u8a66\u884c\u304c\u7d42\u308f\u308b\u307e\u3067\u306b, \u300c\u88cf\u2192\u8868\u300d\u306e\u9806\u3067\u51fa\u306a\u3044\u4e8b\u8c61\u3092 \\(B _ n\\) \u3067\u8868\u3059. \u3064\u304e\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u78ba\u7387 \\(P( A _ n ) , \\ P( B _ n )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u78ba\u7387 \\(P( A _ n \\cap B _ n )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\u6975\u9650\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{P( A _ n ) P( B _ n )}{P( A _ n \\cap B _ n )}\r\n\\]\r\n\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(0 \\lt r \\lt 1\\) \u3092\u307f\u305f\u3059 \\(r\\) \u306b\u5bfe\u3057\u3066, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} nr^n =0\\) \u3068\u306a\u308b\u3053\u3068\u3092\u8a3c\u660e\u306a\u3057\u306b\u7528\u3044\u3066\u3082\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(A _ n\\) \u306f, \u8868\u304c\u51fa\u308b\u56de\u6570\u304c \\(0\\) \u56de, \\(1\\) \u56de\u3067\u3042\u308b\u4e8b\u8c61\u306e\u4f59\u4e8b\u8c61\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nP( A _ n ) & = 1 -(1-a)^n -{} _ {n} \\text{C} {} _ 1 a (1-a)^{n-1} \\\\\r\n& =\\underline{1 -(1-a)^n -na(1-a)^{n-1}}\r\n\\end{align}\\]\r\n\\(B _ n\\) \u306f, \\(k\\) \u56de\u76ee\uff08 \\(0 \\leqq k \\leqq n\\) \uff09\u307e\u3067\u8868\u304c\u51fa\u3066, \u6b8b\u308a\u306f\u88cf\u304c\u51fa\u308b\u4e8b\u8c61\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nP( B _ n ) & = \\textstyle\\sum\\limits _ {k=0}^n a^k (1-a)^{n-k} \\\\\r\n& = \\dfrac{(1-a)^{n+1}-a^{n+1}}{(1-a)-a} \\\\\r\n& = \\underline{\\dfrac{(1-a)^{n+1}-a^{n+1}}{1-2a}}\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\(A _ n \\cap B _ n\\) \u306f, \\(k\\) \u56de\u76ee\uff08 \\(2 \\leqq k \\leqq n\\) \uff09\u307e\u3067\u8868\u304c\u51fa\u3066, \u6b8b\u308a\u306f\u88cf\u304c\u51fa\u308b\u4e8b\u8c61\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nP( A _ n \\cap B _ n ) & = \\textstyle\\sum\\limits _ {k=2}^n a^k (1-a)^{n-k} \\\\\r\n& = a^2 \\textstyle\\sum\\limits _ {k=0}^{n-2} a^k (1-a)^{n-2-k} \\\\\r\n& = \\underline{\\dfrac{a^2 \\left\\{ (1-a)^{n-1}-a^{n-1} \\right\\}}{1-2a}}\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \\[\\begin{align}\r\n\\dfrac{P( A _ n ) P( B _ n )}{P( A _ n \\cap B _ n )} & = \\left\\{ 1 -(1-a)^n -na(1-a)^{n-1} \\right\\} \\\\\r\n& \\qquad \\cdot \\dfrac{(1-a)^{n+1}-a^{n+1}}{a^2 \\left\\{ (1-a)^{n-1}-a^{n-1} \\right\\}} \\\\\r\n& = \\left\\{ 1 -(1-a)^n -\\frac{a}{1-a} \\cdot n(1-a)^{n} \\right\\} \\\\\r\n& \\qquad \\cdot \\dfrac{(1-a)^2- a^2 \\left( \\frac{a}{1-a} \\right)^{n-1}}{a^2 \\left\\{ 1 -\\left( \\frac{a}{1-a} \\right)^{n-1} \\right\\}}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(0 \\lt 1-a \\lt 1\\) , \\(0 \\lt \\frac{a}{1-a} \\lt 1\\) \u306a\u306e\u3067, \\(n \\rightarrow \\infty\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n& (1-a)^n \\rightarrow 0 , \\ n (1-a)^n \\rightarrow 0 , \\\\\r\n& \\left( \\frac{a}{1-a} \\right)^{n-1} \\rightarrow 0\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} & \\dfrac{P( A _ n ) P( B _ n )}{P( A _ n \\cap B _ n )} \\\\\r\n& = \\left\\{ 1 -0 -\\dfrac{a}{1-a} \\cdot 0 \\right\\} \\cdot \\dfrac{(1-a)^2 -a^2 \\cdot 0}{a^2 ( 1-0 )} \\\\\r\n& = \\underline{\\dfrac{(1-a)^2}{a^2}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u8868\u304c\u51fa\u308b\u78ba\u7387 \\(a \\ \\left( 0 \\lt a \\lt \\dfrac{1}{2} \\right)\\) , \u88cf\u304c\u51fa\u308b\u78ba\u7387\u304c \\(1-a\\) \u306e\u30b3\u30a4\u30f3\u3092 \\(1\\) \u679a\u6295\u3052\u308b\u8a66\u884c\u3092 \\(n\\) \u56de\u884c\u3046. \u305f\u3060\u3057 \\( … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n