\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n
A, B, C\u306e3\u4eba\u304c, A, B, C, A, B, C, A, \u30fb\u30fb\u30fb\u306e\u9806\u306b\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052, \u6700\u521d\u306b1\u306e\u76ee\u3092\u51fa\u3057\u305f\u4eba\u304c\u52dd\u3061\u3068\u306a\u308b\u30b2\u30fc\u30e0\u304c\u3042\u308a, \u3060\u308c\u304b\u304c1\u3092\u51fa\u3059\u304b, \u3082\u3057\u304f\u306f, \u5168\u54e1\u304c\\(n\\)\u56de\u305a\u3064\u6295\u3052\u305f\u3089 \u3053\u3061\u3089\u306f, \u5909\u6570\\(n\\)\u304c\u7d61\u3080\u78ba\u7387\u3092\u6c42\u3081\u308b\u3068\u3044\u3046\u3053\u3068\u3067, \u4e00\u898b\u96e3\u3057\u305d\u3046\u306b\u898b\u3048\u307e\u3059\u304c, \u672c\u8cea\u7684\u306b\u306f, \u30b3\u30a4\u30f3\u30c8\u30b9\u3067\\(k\\)\u56de\u76ee\u306b\u521d\u3081\u3066\u8868\u304c\u51fa\u308b\u78ba\u7387, \u3092\u6c42\u3081\u308b\u3088\u3046\u306a\u554f\u984c\u3068\u540c\u3058\u3067\u3059. A\u304c\u52dd\u3064\u3068\u3044\u3046\u4e8b\u8c61\u3092\u6392\u53cd\u306a\u4e8b\u8c61\u306e\u548c\u4e8b\u8c61\u306e\u5f62\u3067\u8868\u3057\u3066\u78ba\u7387\u3092\u8a08\u7b97\u3057\u3066\u3044\u304d\u307e\u3059. \\(A\\)\u3092\u300cA\u304c\u52dd\u3064\u300d\u4e8b\u8c61\u3068\u3057, \\(A_k\\) \\((k=1,2,\\cdots, n)\\)\u3092\u300cA\u304c\\(k\\)\u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u6295\u3052\u3067\u52dd\u3064\u300d\u4e8b\u8c61\u3068\u3059\u308b. \\(B\\), \\(B_k\\), \\(C\\), \\(C_k\\)\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u5b9a\u7fa9\u3059\u308b. \u3044\u304f\u3064\u304b\u88dc\u8db3\u3092\u3057\u307e\u3059. \u307e\u305a, \u6b21\u306b\u3053\u306e\u30b2\u30fc\u30e0\u306f\u8ab0\u3082\u52dd\u305f\u306a\u3044(=\u5f15\u304d\u5206\u3051)\u306e\u72b6\u614b\u3067\u30b2\u30fc\u30e0\u304c\u7d42\u4e86\u3059\u308b\u3053\u3068\u304c\u3042\u308a\u5f97\u307e\u3059. \u305d\u308c\u306fA, B, C\u304c\u5404\\(n\\)\u56de\u4e00\u5ea6\u30821\u306e\u76ee\u3092\u51fa\u305b\u306a\u3044\u3068\u304d\u3067, \u305d\u306e\u78ba\u7387\u306f,$$ \u6c42\u3081\u305f\u78ba\u7387\u304c\u8aa4\u3063\u3066\u3044\u305f\u5834\u5408, \u4e0a\u8a18\u306e\u3088\u3046\u306b\u300c\u76f4\u611f\u306b\u53cd\u3057\u3066\u3044\u306a\u3044\u304b\u300d\u3001\u300c\u5168\u3066\u306e\u5834\u5408\u306e\u78ba\u7387\u3092\u8db3\u3057\u3066\\(1\\)\u306b\u306a\u308b\u304b\u300d\u3092\u78ba\u8a8d\u3059\u308b\u3053\u3068\u3067, \u8a08\u7b97\u9593\u9055\u3044\u306b\u6c17\u3065\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093.<\/p>\n\n\n\n \u307e\u305f, \u7406\u7cfb\u306e\u7bc4\u56f2(\u6570\u2162)\u306e\u88dc\u8db3\u306b\u306a\u308a\u307e\u3059\u304c, \u4eca\u56de\u306e\u30eb\u30fc\u30eb\u3092\u6539\u5909\u3057\u3066, \u300c\\(n\\)\u56de\u305a\u3064\u6295\u3052\u7d42\u308f\u3063\u305f\u3089\u5f15\u304d\u5206\u3051\u300d\u3068\u3044\u3046\u30eb\u30fc\u30eb\u3092\u5916\u3057, \u300c\u6c7a\u7740\u304c\u3064\u304f\u307e\u3067\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u7d9a\u3051\u308b\u300d\u3068\u3057\u3066\u307f\u307e\u3057\u3087\u3046. \u3053\u306e\u3068\u304d, A, B, C\u304c\u52dd\u3064\u78ba\u7387\u306f, \u305d\u308c\u305e\u308c\u4e0a\u3067\u6c42\u3081\u305f\\(P(A)\\), \\(P(B)\\), \\(P(C)\\)\u3067\\(n\\rightarrow \\infty\\)\u3068\u3057\u305f\u3082\u306e\u306b\u306a\u308a\u307e\u3059. $$ \\lim_{n\\rightarrow \\infty} \\left(\\frac{5}{6}\\right)^{3n}=0$$\u3088\u308a, \u305d\u306e\u78ba\u7387\u306f\u305d\u308c\u305e\u308c, \\(\\displaystyle \\frac{36}{91}\\), \\(\\displaystyle \\frac{30}{91}\\), \\(\\displaystyle \\frac{25}{91}\\)\u3068\u306a\u308a, \u3053\u306e\u30eb\u30fc\u30eb\u306e\u5834\u5408\u306f\u5f15\u304d\u5206\u3051\u304c\u306a\u3044\u305f\u3081, \u3053\u308c\u30893\u3064\u306e\u78ba\u7387\u306e\u548c\u306f\\(1\\)\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059. \u3053\u306e\u3088\u3046\u306b, \\(n\\)\u306a\u3069\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u304c\u5165\u3063\u305f\u78ba\u7387\u3067\u305d\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u7121\u9650\u5927\u306b\u3057\u3066, \u305d\u306e\u78ba\u7387\u306e\u610f\u5473\u3092\u8003\u3048\u308b\u306e\u3082\u9762\u767d\u3044\u3067\u3059.<\/p>\n\n\n\n youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n
\u30b2\u30fc\u30e0\u3092\u7d42\u4e86\u3059\u308b. A, B, C\u304c\u52dd\u3064\u78ba\u7387\u3092\u305d\u308c\u305e\u308c\u6c42\u3081\u3088.
(2023 \u4e00\u6a4b\u5927\u5b66 [5])<\/span><\/p>\n\n\n\n
\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n
\u3053\u306e\u3068\u304d, $$
A=A_1 \\cup A_2 \\cup \\cdots \\cup A_n
$$\u3067\u3042\u308a, \u5404\\(A_k\\)\u306f\u4e92\u3044\u306b\u6392\u53cd\u3067\u3042\u308b. \u3088\u3063\u3066, A\u304c\u52dd\u3064\u78ba\u7387\\(P(A)\\)\u306f, $$
P(A)=\\sum_{k=1}^nP(A_k)
$$\u3068\u306a\u308b.
\u3053\u3053\u3067, \u4e8b\u8c61\\(A_k\\)\u304c\u8d77\u3053\u308b\u306e\u306f, A\u304c\\(k\\)\u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u308b\u307e\u3067\u306b, A, B, C\u306e3\u4eba\u304c\u6295\u3052\u305f\u5408\u8a08\\(3(k-1)\\)\u56de\u5168\u30661\u4ee5\u5916\u306e\u76ee\u304c\u51fa\u3066, \u305d\u306e\u5f8cA\u304c\\(k\\)\u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u6295\u3052\u30671\u3092\u51fa\u3059\u78ba\u7387\u306a\u306e\u3067,
$$
P(A_k)=\\left(\\frac{5}{6}\\right)^{3(k-1)}\\cdot\\frac{1}{6}
$$\u3068\u306a\u308b. \u3088\u3063\u3066, $$
\\begin{align}
P(A)&=\\sum_{k=1}^nP(A_k)\\\\[1.5ex]
&=\\sum_{k=1}^n\\left\\{\\left(\\frac{5}{6}\\right)^{3(k-1)}\\cdot\\frac{1}{6}\\right\\}\\\\[1.5ex]
&=\\frac{1}{6}\\sum_{k=1}^n\\left\\{\\left(\\frac{5}{6}\\right)^3\\right\\}^{k-1}\\\\[1.5ex]
&=\\frac{1}{6}\\cdot\\frac{1-\\left(\\frac{5}{6}\\right)^{3n}}{1-\\left(\\frac{5}{6}\\right)^3}\\\\[1.5ex]
&=\\frac{36}{91}\\left\\{1-\\left(\\frac{5}{6}\\right)^{3n}\\right\\}
\\end{align}
$$\u3068\u6c42\u307e\u308b. \u3053\u3053\u3067, \\(\\displaystyle \\sum_{k=1}^n\\left\\{\\left(\\frac{5}{6}\\right)^3\\right\\}^{k-1}\\)\u306f, \u521d\u9805\\(1\\), \u516c\u6bd4\\(\\displaystyle \\left(\\frac{5}{6}\\right)^3\\), \u9805\u6570\\(n\\)\u3068\u3057\u3066, \u7b49\u6bd4\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u6c42\u3081\u305f. \\(B\\)\u3082\u540c\u69d8\u306b,
$$
P(B)=\\sum_{k=1}^nP(B_k)
$$\u3068\u306a\u308a, \u4e8b\u8c61\\(B_k\\)\u304c\u8d77\u3053\u308b\u306e\u306f, B\u304c\\(k\\)\u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u308b\u307e\u3067\u306b, A, B, C\u306e3\u4eba\u304c\u6295\u3052\u305f\u5408\u8a08\\(3k-2\\)\u56de\u5168\u30661\u4ee5\u5916\u306e\u76ee\u304c\u51fa\u3066, \u305d\u306e\u5f8cB\u304c\\(k\\)\u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u6295\u3052\u30671\u3092\u51fa\u3059\u78ba\u7387\u306a\u306e\u3067,
$$
P(B_k)=\\left(\\frac{5}{6}\\right)^{3k-2}\\cdot\\frac{1}{6}
$$\u3068\u306a\u308b. \u3088\u3063\u3066, $$
\\begin{align}
P(B)&=\\sum_{k=1}^nP(B_k)\\\\[1.5ex]
&=\\sum_{k=1}^n\\left\\{\\left(\\frac{5}{6}\\right)^{3k-2}\\cdot\\frac{1}{6}\\right\\}\\\\[1.5ex]
&=\\frac{5}{6}\\sum_{k=1}^n\\left\\{\\left(\\frac{5}{6}\\right)^{3(k-1)}\\cdot\\frac{1}{6}\\right\\}\\\\[1.5ex]
&=\\frac{5}{6}\\times P(A)\\\\[1.5ex]
&=\\frac{30}{91}\\left\\{1-\\left(\\frac{5}{6}\\right)^{3n}\\right\\}
\\end{align}
$$\u3068\u306a\u308b. \u6700\u5f8c\u306b, \\(P(C)\\)\u3082\u540c\u69d8\u306b\u6c42\u3081\u308b. \u4e8b\u8c61\\(C_k\\)\u304c\u8d77\u3053\u308b\u306e\u306f, C\u304c\\(k\\)\u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u308b\u307e\u3067\u306b, A, B, C\u306e3\u4eba\u304c\u6295\u3052\u305f\u5408\u8a08\\(3k-1\\)\u56de\u5168\u30661\u4ee5\u5916\u306e\u76ee\u304c\u51fa\u3066, \u305d\u306e\u5f8cC\u304c\\(k\\)\u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u6295\u3052\u30671\u3092\u51fa\u3059\u78ba\u7387\u306a\u306e\u3067,$$
P(C_k)=\\left(\\frac{5}{6}\\right)^{3k-1}\\cdot\\frac{1}{6}
$$\u3067\u3042\u308a,
$$
\\begin{align}
P(C)&=\\sum_{k=1}^nP(C_k)\\\\[1.5ex]
&=\\sum_{k=1}^n\\left\\{\\left(\\frac{5}{6}\\right)^{3k-1}\\cdot\\frac{1}{6}\\right\\}\\\\[1.5ex]
&=\\left(\\frac{5}{6}\\right)^2\\sum_{k=1}^n\\left\\{\\left(\\frac{5}{6}\\right)^{3(k-1)}\\cdot\\frac{1}{6}\\right\\}\\\\[1.5ex]
&=\\left(\\frac{5}{6}\\right)^2\\times P(A)\\\\[1.5ex]
&=\\frac{25}{91}\\left\\{1-\\left(\\frac{5}{6}\\right)^{3n}\\right\\}
\\end{align}
$$\u3068\u6c42\u307e\u308b.<\/p>\n\n\n\n
$$
P(A)>\\frac{5}{6}P(A)=P(B)>\\left(\\frac{5}{6}\\right)^2P(A)=P(C)
$$\u304b\u3089, \u30b2\u30fc\u30e0\u306b\u52dd\u3064\u78ba\u7387\u306fA\u304c\u6700\u3082\u5927\u304d\u304f, C\u304c\u6700\u3082\u5c0f\u3055\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059. \u3053\u308c\u306f\u76f4\u611f\u901a\u308a\u3067, \u5148\u306b\u6295\u3052\u308b\u3053\u3068\u304c\u3067\u304d\u308bA\u304c, \u4e00\u756a\u52dd\u3064\u78ba\u7387\u304c\u5927\u304d\u3044\u306f\u305a\u3067\u3059. A\u304c1\u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u6295\u3052\u30671\u306e\u76ee\u3092\u51fa\u3059\u3068, B, C\u306f\u30b5\u30a4\u30b3\u30ed\u30921\u5ea6\u3082\u6295\u3052\u308b\u3053\u3068\u306a\u304f, \u30b2\u30fc\u30e0\u306f\u7d42\u4e86\u3057\u3066\u3057\u307e\u3044\u307e\u3059. C\u304c\u52dd\u3064\u78ba\u7387\u304c\u4e00\u756a\u5c0f\u3055\u3044\u306e\u3082\u306e\u76f4\u611f\u901a\u308a\u3067\u3059.<\/p>\n\n\n\n
P(\u5f15\u304d\u5206\u3051)=\\left(\\frac{5}{6}\\right)^{3n}
$$\u3068\u306a\u308a\u307e\u3059. \u30b2\u30fc\u30e0\u7d42\u4e86\u6642\u70b9\u3067, \u300cA\u304c\u52dd\u3064\u300d, \u300cB\u304c\u52dd\u3064\u300d, \u300cC\u304c\u52dd\u3064\u300d, \u300c\u5f15\u304d\u5206\u3051\u300d\u306e\u3044\u305a\u308c\u304b1\u3064\u304c\u5fc5\u305a\u8d77\u304d, \u305d\u3057\u3066\u3069\u306e2\u3064\u3082\u540c\u6642\u306b\u8d77\u3053\u308a\u5f97\u306a\u3044\u305f\u3081,$$
P(A)+P(B)+P(C)+P(\u5f15\u304d\u5206\u3051)=1
$$\u3068\u306a\u308b\u306f\u305a\u3067\u3059\u3057, \u4eca\u56de\u6c42\u3081\u305f\u5404\u78ba\u7387\u3092\u5165\u308c\u308b\u3068\u5b9f\u969b\u306b\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059.<\/p>\n\n\n\n