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求一个整数的所有素数因子的思路是什么？ | 1. 一般的小数可以用简单筛法找出质数列表，然后一个个试。这种方法简单暴力，但是对几亿以下的数字可以很快。
2\. 再大一点的数 $$n$$ 就用Pollard的 $$\rho$$ 算法，思路：
任取一个数 $$x_0$$ 开始，不断计算 $$x_k=(x_{k-1}^2+1) \text{ mod } n$$ ，则如果 $$n$$ 有质因数$$p$$ ，那么 $$x_k \text{ mod } p$$ 应该能更快地进入循环，可以用龟兔赛跑算法（图形状像 $$\rho$$，因此算法得名）试图找出这个循环点，一旦找到 $$x_k\equiv x_l \text{ mod } p$$ ，立刻可以算 $$x_k-x_l$$ 和$$n$$ 的最大公约数得到分解。有一定几率失败。
3\. 更大的但在 $$10^{25}$$ 以下的 $$n$$ 可以用Lenstra椭圆曲线算法（ECM），思路：
挑选椭圆曲线 $$y^2=(x^3+ax+b) \text{ mod } n$$ 外加上面某一点 $$P$$ ，然后取一个较小的阶乘$$n!$$，反复用椭圆曲线加法算出 $$2!P, 3!P, \cdots, n!P$$ ，在重复计算加法中每一步都会计算和曲线切线斜率 $$x/y$$同余的整数，即找出整数 $$z$$ 使得$$yz\equiv x \text{ mod } n$$ ，方法为找 $$y$$ 和 $$n$$的最大公约数，在这个过程中如果得出公约数大于1则分解成立，有一定几率失败。
4\. 更大的 $$10^{100}$$ 以下的 $$n$$ 可以用二次筛（QS），思路
费尔马分解法试图把奇数 $$n$$ 写成 $$n=a^2-b^2$$ 的方法，这样立刻可得因数分解 $$(a+b)(a-b)$$，二次筛就是一种可以高效找到此类数字的方法，试图测试多个介于 $$\sqrt n$$ 和 $$n-1$$ 之间的数 $$a$$ ，如果 $$a^2\text{ mod } n$$ 正好是完全平方那就成了，不然找到几个不同的 $$a_i^2 \text{ mod } n$$，如果乘积正好是完全平方数，那么 $$a=\prod_i a_i \text{ mod } n$$ 满足条件。
5\. 再往上只能用普通数域筛选法（GNFS），这个就很复杂了，思路
也是用费尔马分解法，只是比二次筛更高效，核心定理：
取系数为整数的 $$d$$ 次多项式 $$f(x)$$ 有复根 $$\theta$$ ，并存在不是 $$n$$ 倍数的整数 $$m$$ ，使得$$f(m)$$ 为$$n$$ 倍数，将所有次数不超过 $$d-1$$的整系数多项式代入$$\theta$$后的值定义为一个“环”（因为任何两个这样形式的数的和或积依然是这样形式的数），那么必然存在一个将这个环里的复数映射到不是$$n$$ 倍数的整数上的函数 $$\phi(x)$$ 满足：
i - $$\phi(ab)=\phi(a)\phi(b)$$
ii - $$\phi(a+b)=\phi(a)+\phi(b)$$
iii - $$\phi(1)\equiv1 \text{ mod } n$$
iv - $$\phi(\theta) \equiv m \text{ mod } n$$
已知这个定理后，如果找到整数对 $$(a_i,b_i)$$ 使得复数 $$\beta = \prod_i (a_i+b_i\theta)$$ ，整数 $$y= \prod_i (a_i+b_im)$$ ，以及 $$\beta$$ 的映像 $$x=\phi(\beta)$$ ，则很容易根据i-iv得到$$x^2\equiv y^2 \text{ mod } n$$ ，有2/3机会能由此找到 $$n $$ 的一个约数。
2009年，232位的RSA-768数通过上百台计算机耗时两年成功分解，用的就是高度优化后的GNFS。 | {
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|
两个人连续投一枚硬币，如果先出现“正正反”则甲胜，如果先出现“正反反”则乙胜，两人获胜概率是否一样？假设硬币质地均匀，每次投硬币正反两面概率都是50%。连续投很多次，直到出现“正正反”或“正反反”为止。 | 第一次如果投反，那么甲乙都等于推倒重练，两人获胜概率不会有任何变化。
所以现在只需要考虑第一次投正的情况，假设此时甲胜率为 $$x$$ ，乙为 $$y$$ ，那么有两种概率各位1/2的情况：
1 - 第二次投出正，这时甲将100%获胜，因为这一刻开始接下来除非不断投出正，只要一出现反甲就赢了。
2 - 第二次投出反，这时看第三投，有1/2概率乙胜利（正反反），剩下1/2概率出现正反正，对于甲乙来说胜率相当于第一次投出正的情况，甲为 $$x$$，乙为 $$y$$。
综合
$$x=\frac{1}{2}\cdot1+\frac{1}{2}\left(\frac{1}{2}\cdot0+\frac{1}{2}x\right)$$ ，得出$$x=\frac{2}{3}$$
或者
$$y=\frac{1}{2}\cdot0+\frac{1}{2}\left(\frac{1}{2}\cdot1+\frac{1}{2}y\right)$$ ，得出$$y=\frac{1}{3}$$
两者相加为1，验算通过。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:584880907, aid:2904885349 | human | true | 暂无版权及作者信息 | [
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|
数论的威尔逊定理已知有哪些证明方法？主要是想问必要性的证明，目前找到的有基于原根逆元的证法，有Lagrange证法，还有哪些证明方法？ | 威尔逊有三种常见基础证明：
1. 标准教科书证明（稍微拓宽到一般情况），可能就是题主所说的Lagrange证法
设 $$p$$ 为奇素数（ $$p=2$$ 明显），设 $$D$$ 与 $$p$$ 互质，讨论同余方程：
$$xy\equiv D\bmod p$$
设 $$x,y\in\\{1,2,3,\cdots,p-1\\}$$ ，因为 $$x,y$$ 与 $$p$$ 互质，对于每一个 $$x$$必然存在一个唯一的 $$y$$ 满足以上方程（不然 $$x(y_1-y_2)$$ 为 $$p$$ 的倍数，矛盾）。
如果存在 $$x^2\equiv D\bmod p$$ （即 $$D$$ 为 $$p$$ 的二次剩余），那么根据 $$x^2-y^2=(x-y)(x+y)\equiv 0\bmod p$$ ，可得 $$y=p-x$$ ，即除了 $$x,p-x$$ 外的数都不可能满足$$x^2\equiv D\bmod p$$ 。
所以 $$\\{1,2,3,\cdots,p-1\\}$$ 中除 $$x,p-x$$ 外的数可以两两配对满足 $$xy\equiv D\bmod p$$，而 $$x(p-x)=px-x^2\equiv -x^2\bmod p\equiv-D\bmod p$$
因此 $$(p-1)!\equiv -D^{\frac{p-1}{2}}\bmod p$$
代入 $$D=1$$ 就得到威尔逊定理。
此外，代入 $$(p-1)!\equiv -1\bmod p$$ 可得 $$D^{\frac{p-1}{2}}\equiv 1\bmod p$$ ，如果$$D$$ 是 $$p$$ 的二次非剩余根据以上步骤可得 $$D^{\frac{p-1}{2}}\equiv -1 \bmod p$$ ，由此得到$$D^{\frac{p-1}{2}}\equiv\pm 1 \bmod p$$ 可以用来判定 $$D$$ 是否为 $$p$$的二次剩余，即鼎鼎大名的欧拉判定，由此可进一步推导如平方互反等重要结论。
2. 原根
取 $$p$$ 的原根 $$g$$ ，则 $$1,g,g^2,\cdots,g^{p-2}$$ 除以 $$p$$ 的余数为$$1,2,3,\cdots,p-1$$ 。则
$$(p-1)!=g^{0+1+2+\cdots+(p-2)}=g^{\frac{(p-2)(p-1)}{2}}\bmod p$$
设 $$p=2k+1$$ ，则 $$(p-1)!\equiv g^{k(2k-1)}\bmod p$$
根据费尔马小定律，已知 $$(g^k)^2=g^{2k}=g^{p-1}\equiv 1\bmod p$$ ，则 $$g^k\equiv \pm 1\bmod p$$ ，但 $$k<p$$ ，因此 $$g^k\not\equiv1\bmod p$$ ，所以只能 $$g^k\equiv -1 \bmodp$$ ，那么
$$(p-1)!=g^{k(2k-1)}=(-1)^{2k-1}=-1\bmod p$$
3. 既然用了费尔马小定律，有一个代数证明：
考虑多项式
$$f(x)=x^{p-1}-1-(x-1)(x-2)\cdots(x-p+1)$$
根据小费， $$f(1)=f(2)=\cdots=f(p-1)\equiv 0\bmod p$$
因此， $$1,2,\cdots,p-1$$ 是 $$p-2$$ 次多项式（最高项消除） $$f(x)$$ 模 $$p$$ 的根，因此可得$$f(x)\equiv 0 \bmod p$$ ，所有系数皆为 $$p$$ 的倍数，包括常数项，即
$$-1-(-1)^{p-1}(p-1)!\equiv 0 \bmod p$$
$$(p-1)!\equiv -1\bmod p$$
如果用群论的话，可以直接将西罗定理3用在对称群上得到。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:558886947, aid:2714940196 | human | true | 暂无版权及作者信息 | [
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|
有没有什么量级达到了10^-(10^2)? | 这个数就是1古戈尔分之一（另，这就是谷歌公司名字的来源）。
在数学上当然这个数不算什么大数。直观来说70的阶乘差不多是这个数量级的。换句话说，如果有70个人排队进场，你想要得到一个特定顺序的概率就是$$10^{-100}$$ 。
从物理上来说，这个量级还是很大的。举几个例子：
1. 一个电子的质量差不多是可观测宇宙质量的 $$10^{-80}$$ 到 $$10^{-90}$$ 级别，这个差距是题主所说的数的至少100亿倍。
2. 2022年超级计算机世界第一是美国的前沿计算机，运算速度超过了100万TFLOP（即每秒可以运行的浮点运算次数达到 $$10^{18}$$次级别）。整台计算机占地680平方米，和一个篮球场差不多大，造价约6亿美元。如果把这台计算机缩小到1个原子那么大，然后把整个可观测宇宙里的原子都变成前沿，那么所有这些计算机并行100秒可以一加一一直加到1古戈尔。
3. 宇宙内的基本粒子总数估计为 $$10^{80}$$ 个。如果全宇宙都塞满中子估计共可塞下 $$10^{128}$$个。阿基米德当年计算宇宙中可以塞下多少沙粒得到了 $$10^{63}$$ 的数字，但是他估计中的宇宙直径只有2光年，如果换成可观测宇宙则能塞下的总沙粒数为$$10^{95}$$ 。所以 $$10^{-100}$$ 相当于沙粒体积和1万个可观测宇宙体积的比例。
那么有没有什么量级能达到这么大呢？星系质量级别的超大黑洞（ $$10^{11}$$ 个太阳质量）通过霍金辐射逐渐蒸发殆尽所需的时间差不多是$$10^{100}$$ 年等级。如果热寂说是正确的，这可以被视为其发生时间点的下限。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:613494313, aid:3132435413 | human | true | 暂无版权及作者信息 | [
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|
身材和颜值是完全独立的还是有相关性的？有没有相关研究和结论 | 2014年发表在《性行为档案》期刊中的一篇论文
Bleske-Rechek A, Kolb CM, Stern AS, Quigley K, Nelson LA. Face and body:independent predictors of women's attractiveness. Arch Sex Behav. 2014Oct;43(7):1355-65. doi: 10.1007/s10508-014-0304-4. Epub 2014 May 15. PMID:24830907.
该文章将女性照片分为只有脸部，只有身体和全身，并包括日常衣着和泳装两组让评判者（男女都有）进行打分。受试人是日常生活中18-24岁的女大学生。显然，这里的结论只适用于美国人审美。
以下为一些结论：
1. 脸部吸引力和身体吸引力存在相当的相关性
这和之前的类似研究相符，但是发现的相关性更强。
在日常衣着组，脸和身体的吸引力有0.62的相关性。在泳装组，这个相关性降为0.45。可以看出当目标穿泳衣时，身材吸引力出现更大的两极分化（好坏更为清楚），因此降低了其和颜值的相关性。
此外，论文发现两者对整体吸引力有独立贡献。这一点很好理解，即使身材完全一样，颜值增加也会增加整体吸引力，反之亦然。
2. 身体露得越多，身材对整体吸引力的影响越不受颜值影响
在日常衣着组，身体吸引力和整体吸引力相关性为0.69，但是控制颜值后这一相关性降为0.39。
在泳装组，身体吸引力和整体吸引力相关性高达0.82，而且控制颜值之后并没有太大影响（0.81）。
3. 其它身材指数都和身体吸引力有相关性
首先，和预计的一样，腰臀比和身体质量指数有相关性（0.44到0.48）。两者都和身体吸引力成负相关，即腰臀比和身体质量指数越大，身体吸引力越低。在日常衣着组，腰臀比和身体吸引力相关性为-0.27，身体质量指数为-0.68。在泳装组，两者相关性显著增加，分别为-0.58和-0.76。
这说明相对腰臀比，是否过重（身体质量指数）对身材扣分更大，而且腰臀比和着装关联不大，但是身体质量指数在穿得少时负面影响显著增加。
此外，研究发现在日常衣着组，胸围与各项吸引力没有什么相关性，罩杯和身体以及脸的吸引力也没有相关性，但是和整体吸引力负相关。在泳装组，胸围和罩杯与所有吸引力都呈清楚的负相关。
以上结论可能反直觉，但是这也标志在日常人群中，苗条且前凸后翘出现的可能性极低，胸围和罩杯与体重以及腰臀比有相当的相关性。
4. 身材不止于腰臀比，体重和胸
研究发现即使控制了颜值，腰臀比，身体质量指数和胸大小后，身体吸引力和整体吸引力的相关性并没有太多影响，从单纯只控制颜值的情况下从0.39降到0.34（日常衣着），从0.81下降到0.71（泳装）。这说明身体吸引力不止于身形，体重和胸那么简单，还存在其它因素。 | {
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] | qid:599260366, aid:3016367224 | human | true | 暂无版权及作者信息 | [
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人类为什么偏好数字7？7这个数字在生活中很常见。1/7循环小数的循环节142857在分数中很常见。7是个素数。让人类随便说一个数字，7是最大概率的 | 数字偏好和文化有很深的渊源。我知道在西方文化圈里7是很重要的一个数，但是我并不觉得同样的情况发生在东亚文化圈内。
只谈7在西方文化中的重要性：
## 根本原因：7 是圣数
7在基督教文化中是一个圣数，有7罪7德，地狱9层但天国7层，详见耶鲁大学一位教授在1858年写的手稿《数字七》。基督教为什么把7奉为神圣的道理是很简单的，因为在创世纪里神以6天创造世界，1天休息，所以一周有七天，7变成了一个神圣的数字。
这种宗教上的原因很快渗透了生活的方方面面。就和中国人重五行所以世间万物都会按五之数划分，西方社会也开始以七划分平时见到的事物。实际上，人的脑洞是无限大的，只要想好一个数字总是能硬凑出来。
于是出现了7大洲（欧亚大陆硬是被一分为二），7大洋（硬分四大洋），7色彩虹（如果不先入为主你觉得彩虹几色？），古代人选奇迹也要选7大（当然这是基督教之前的，因为7的意义其实更本质，下面会提到）。
此外英语中也出现了7是幸运数字这一说法，最终也来源于这一“神圣”涵义。
但是，为什么圣经要说世界在7天里创造呢？因为早在基督教建立之前，7天的星期概念已经在巴比伦、波斯、罗马出现了。和很多其它事物一样，基督教建立后重新解读了很多古代习俗（例如用圣诞节覆盖传统上的冬至庆典和农神节等等），在教义内重新定义。
那为什么古巴比伦会用七天为一周呢？根源在于天空中行走的七星：金木水火土日月，那是远古时代天空中肉眼可见会动的仅有天体，在很多古文明中都有巨大影响。星期和七星的联系在英语中因为受到北欧神祇名字的影响不是很明显（Monday-月,Sunday-日, Saturday-土星还是保留了联系），但是在拉丁语系中都保留了下来。日本西化后直接借用七星来命名一周七天，就连中文的“星期”一词也是这么来的。
## 其它偏门的理由
1. 英文中7是最朗朗上口的一个数词，Seven是一位数中唯一有两个音节的词，所以放在诗歌韵律中使用特别活。Snow White and theEight Dwarfs 听上去就比 Snow White and the Seven Dwarfs拗口。
2. 心理学家乔治·米勒曾提出人大脑的短期记忆可以容纳的事物在5-9之间，所以常有“最多同时记7件事”，“背诵时要把对象分为7个一组记忆”这些说法。
3. 7在个位数里是最特异的，它是10以下最大的质数，要判断一个数是2、3、5的倍数都很简单，但是7的倍数就会更费周章。所以你让人从1-10里挑一个数，人们会下意识跳开直觉“有规律”的数，例如偶数，1，5，3和3的倍数，所以“不零不整”的7被挑中的机会最大。
4. 对赌徒来说，7幸运还有一个原因-两个骰子加起来点数为7的概率最大，当然这和7没有半毛钱关系，这是因为骰子正好有6面，那要细说就要涉及到三维正多面体一共只有5种了。 | {
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|
如何理解四维空间的人可以一眼看到一个平面的正反两面？ | 先降一维理解一下，在平面上的生物只能看到直线的一边，这是因为在平面里和直线垂直的方向只有两个（过直线上一点只能作一条直线和该直线垂直，这条“法线”有两个方向），这个生物的视角方向和这两个方向之间的一个形成一个钝角，可以定义为该视角只能看到这个方向“那一边”。
一旦来到三维，直线就失去了某一边的概念，因为过直线上一点可以作无数条和该直线垂直的直线，即这条直线有无数个法线方向，你可以说直线有无数边。三维空间里的一个固定视角方向依然会和这其中某些法线方向成锐角，某些成钝角，如果沿用上面的定义，那么三维空间的我们能够看到这无数边中的一些，但不能看到另一些。特别是原有平面上的两个方向在三维视角下依然有一个是“看得到”，另一个“看不到”，并没有两边都能看到一说。
同理，在四维空间中一个平面有无数个法线方向，即不再有正反两面而是有无数面。四维生物可以看到其中的无数面，但是依然不可能同时看到我们世界里该平面的正反两面。
例如，如果一个平面在三维里的法线方向为 $$(n_x,n_y,n_z)$$ ，那么在四维空间里任意 $$(n_x,n_y,n_z,t)$$都和该平面垂直，沿着这个方向都可以看到平面的一个面，这样的面有无数个。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:606084457, aid:3083028179 | human | true | 暂无版权及作者信息 | [
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|
这道极其复杂集合card题有没有什么好的解法？
已知函数 $f(x)$ 满足 $\left\{x \mid f(x)>f\left(\frac{1}{1-x}\right)=(a, b)(1<a<b)\right.$, 记 $f(x)$ 的最小值为 $M$, 最大值为 $N, S=\{x \mid f(x)=M\}, T=\{x \mid f(x)=N\}$, 则下列命题为假命题的是 (注: $\operatorname{card}(A)$ 表示集合 $A$ 中元素的个数)
A. 若 $\operatorname{card}(S)=1$, 则 $S \subseteq(a, b)$
B. 若 $\operatorname{card}(T)=1$, 则 $T \subseteq(a, b) \quad$
C. 若 $\operatorname{card}(S)=2$, 则 $S \subseteq(a, b)$
D. 若 $\operatorname{card}(T)=2$, 则 $T \subseteq(a, b)$ | 首先注意函数 $$\frac{1}{1-x}$$ 套两次会回到自己
$$\frac{1}{1-\frac{1}{1-x}}=1-\frac{1}{x}$$ ，$$\frac{1}{1-(1-\frac{1}{x})}=x$$
而且当 $$x>1$$ 时， $$\frac{1}{1-x}<0$$ ， $$0<1-\frac{1}{x}<1$$ ，因此这个函数每套用一次，就会在$$(1,\infty)\rightarrow(-\infty, 0)\rightarrow(0,1)$$ 之间循环。
* * *
所以问题的第一个条件就是说，当 $$a<x<b$$ 时
$$f(x)>f(\frac{1}{1-x})\leq f(1-\frac{1}{x})\leq f(x)$$
当 $$x\geq b$$ 或 $$1<x\leq a$$ 时
$$f(x)\leq f(\frac{1}{1-x})\leq f(1-\frac{1}{x})\leq f(x)$$ ，即$$f(x)=f(\frac{1}{1-x})=f(1-\frac{1}{x})$$
* * *
设 $$O=(a,b)\bigcup(\frac{1}{1-a},\frac{1}{1-b})\bigcup(1-\frac{1}{b},1-\frac{1}{a})$$
可见当 $$x\notin O$$ 且 $$x\neq 0,1$$ 时，你有$$f(x)=f(\frac{1}{1-x})=f(1-\frac{1}{x})$$
当 $$x\in O$$ 时，在 $$x,\frac{1}{1-x},1-\frac{1}{x}$$ 三个数中有一个 $$\in (a,b)$$，不失一般性，设 $$x\in (a,b)$$ ，那么 $$f(x)> f(1-\frac{1}{x})\geq f(\frac{1}{1-x})$$ 或者$$f(x)\geq f(1-\frac{1}{x})> f(\frac{1}{1-x})$$
* * *
因此A和C肯定是错的，最小值永远不可能取在 $$x\in(a,b)$$ 内，根据定义 $$f(x)>f(\frac{1}{1-x})$$。最大值如果有2个的话，可能发生一种情况：即 $$x\in (a,b)$$ 和 $$1-\frac{1}{x}\notin (a,b)$$ ，且$$f(x)=f(1-\frac{1}{x})$$ 取到最大值，因此D是错误的。
唯一正确的是B。
* * *
以上都没有考虑0和1，事实上很容易构造满足条件且 $$S=\\{0\\}$$或者 $$S=\\{0,1\\}$$ ， $$T=\\{1\\}$$ 或者$$T=\\{0,1\\}$$ 的情况，这种情况下就连B也可以是错的。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:607407411, aid:3129542311 | human | true | 暂无版权及作者信息 | [
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|
可不可以理解为所有事件的概率都是二分之一？可以不可以把一个事件最后的结果都翻译成，是与不是，能与不能，会或不会，这样所有事件发生的概率都是二分之一。 | 首先看“所有事件的概率是二分之一”
从数学角度来说：
概率的数学表达为概率空间，一个概率空间由三样东西组成：样本空间、事件空间和概率函数。
在有限样本空间的简单情况下用人话说就是有各种基本情况，以及一个会把这些情况的排列组合映射到一个0到1之间的数字的函数。
例如，把骰子掷出6和骰子掷不到6看成两个基本情况，并约定概率函数把两种情况都对应于1/2。这是一个完备的概率空间，事实上，你很容易想象存在这样的骰子。
但是概率函数的构造要满足一些基本规则。
在上面的例子里，“掷不到6“的情况包括掷到1，掷到2，掷到3，掷到4，掷到5这些情况。如果你现在也要讨论这些事件发生的概率必须把它们加入概率空间（不然概率讨论就没有意义了），一旦这么做，根据概率函数的定义你会立刻得到“掷不到6“的概率为掷到1，掷到2，掷到3，掷到4，掷到5的概率总和，那么显然这5个概率不可能都是1/2和依然是1/2。
再看“某一件事的概率是二分之一”
这其实没什么毛病，我们说硬币正面向上概率1/2或骰子掷出6概率为1/6并不是基于任何概率论，而是基于对事件底层物理学的假设。所以说到底这也只是一种假设。
例如看“我彩票中奖的概率为二分之一”，这取决于概率的现实意义。
如果你认为自己买彩票的和别人买彩票的机制完全一致，那么买彩票就是一个发生了千百万次的事件，从概率两大学派之一频率学派的角度来看，你可以观察到所有买过这个彩票的人里面有多少中了奖，从而估计你自己彩票中奖的概率，这显然远远小于二分之一。
如果你认为自己买彩票的和别人买彩票的机制并不一致，存在某种无法观察到的属性影响开奖几率（例如人品），那么如果你之前没买过彩票，这就是一个从未验证过是否会发生的事件，你完全可以假设自己彩票中奖的概率为二分之一。这里可以用第二个学派贝叶斯学派的观点，并没有任何逻辑否认你的假设。但是在这种情况下二分之一这个概率的意义也就只是如此，只是一种“信仰”。然后你去买彩票，每次不中（或者中，不影响结果）之后你会逐渐利用这些新信息更新你自己的“信仰”，最终你对于这件事的概率信仰会越来越低，趋近于上面第一种情况（假设人品属性实际不存在），或者趋近于另一个真实值。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:575605415, aid:2827138204 | human | true | 暂无版权及作者信息 | [
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|
一瓶啤酒要一块钱，两个啤酒瓶能换一瓶啤酒，三个啤酒瓶盖也能换一瓶啤酒。一个人有10块钱，问：不能赊帐，理论上他最多能喝几瓶酒。 | 如果把酒看作是一个连续的量，那么1个酒瓶可以换1/2瓶酒，1个酒盖可以换1/3瓶酒，所以1瓶酒可以换1/2+1/3=5/6瓶酒，这5/6瓶酒又可以换5/6x5/6瓶酒，以此类推，1瓶酒最终可以化为
$$1+\frac{5}{6}+\left(\frac{5}{6}\right)^2+\cdots=\frac{1}{1-\frac{5}{6}}=6$$瓶酒。
当然，实际没有那么理想，比如最后剩下一个瓶盖你就不能再换酒了，以上是最后没有剩下任何酒瓶或酒盖情况下的结果，此时所有酒瓶和酒盖都“转化”了没有浪费。这个答案显然也是原题答案的上限，我们知道原题答案无论如何都不可能超过$$6n$$ 。而且我们知道理想状况下1个酒瓶可以换6x1/2=3瓶酒，1个酒盖可以换6x1/3=2瓶酒。
现在重复以下步骤：
把现有全部酒瓶和酒盖尽可能多全部换成新酒，喝光，然后收集酒瓶和酒盖。
如果剩下的总酒瓶不足2个且总酒盖不足3个，停止，不然重复以上步骤。
可见
1 - 每一步结束时至少有1个酒瓶和1个酒盖。
明显，每一步都会换至少1瓶新酒，所以每一步结束都会有至少1个酒瓶和1个酒盖。
2 - 最后剩下1个酒瓶
如1，如果剩下2个或以上还能再换
3 - 最后剩下2个酒盖
如果某一步开始时，酒盖数为 $$3k+2\geq 2$$ ，则换完之后剩至少2个酒盖。
如果某一步开始时，酒盖数为 $$3k+1\geq 4$$ ，则酒盖能换至少1瓶新酒，加上剩下多余的1个酒盖，总酒盖数至少2。
如果某一步开始时，酒盖数为 $$3k\geq 6$$ ，则酒盖能换至少2瓶新酒，得到2个酒盖。
如果某一步开始时，酒盖数为 $$3$$，则光用酒盖能换1瓶新酒，得到1个酒盖，根据1开始时你至少有1个空酒瓶，那么你此时总共有至少2个空酒瓶，循环不会停止，下一步一开始用2个空酒瓶可以再换1瓶新酒，酒盖数增加为2。
所以最后停下来时你必然剩下1个酒瓶，2个酒盖，折合为“浪费”了7瓶酒的量，一共喝 $$6n-7$$ 瓶酒。
因此，如果这个人有10块钱，则他最终能喝啤酒瓶数为$$6n-7 = 6 imes 10 -7 = 53$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:553387768, aid:2672588607 | human | true | 暂无版权及作者信息 | [
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