Khan Academy - Statistics and Probability - Unit 8 COUNTING, PERMUTATIONS, AND COMBINATIONS

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本文介绍了计数原理，包括阶乘和加法、乘法原理，并深入探讨了排列、组合的概念及其区别。此外，文章讲解了伯努利试验、条件概率、全概率公式和贝叶斯定理，通过实例展示了这些概念在概率问题中的应用。

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COUNTING, PERMUTATIONS, AND COMBINATIONS

PART 1 Counting principle and factorial (计数原理与阶乘)

Part 2 Permutations (排列)

Part 3 Combinations (组合)

Part 4 Combinatorics and probability

Part 5 Conditional probability vs. Total probability vs. Bayesian probability

PART 1 Counting principle and factorial (计数原理与阶乘)

1. Factorial: the factorial of a positive integer $n$ , denoted as $n!$ , is the product of all positive integers less than or equal to $n$ .

(1) $n!=n\times(n-1)\times(n-2)...\times1$

(2) $0!=1$

2. Counting principle: there are two basic principles of counting:

(1) Addition principle: if one event can occur in $m$ ways and a second event with no common outcomes can occur in $n$ ways, then the first or second event can occur in $m+n$ ways.

【加法原理】如果一个目标可以在 $n$ 种不同情况下完成，第 $k$ 种情况又有 $m_k$ 种不同方式来实现，那么实现这个目标总共有 $N=\sum_{i=1}^nm_k=m_1+m_2+...+m_n$ 种方法。

【注意事项】

每种方式都能实现目标，不依赖于其他条件
每种情况内任两种方式都不同时存在
不同情况之间没有相同方式存在

[EXAMPLE] There are 2 vegetarian entrée options and 5 meat entrée options on a dinner menu. What is the total number of entrée options?

[ANSWER] 5+2=7

(2) Multiplication principle: if one event can occur in $m$ ways and a second event can occur in $n$ ways after the first event has occurred, then the two events can occur in $m\times n$ ways. This is also known as the Fundamental Counting Principle.

【乘法原理】如果实现一个目标必须经过 $n$ 个步骤，第 $k$ 步又可以有 $m_k$ 种不同方式来实现，那么实现这个目标总共有种 $N=\Pi_{i=1}^nm_k=m_1m_2...m_n$ 方法。

【注意事项】

步骤可以分出先后顺序，每一步骤对实现目标是必不可少的；
每步的方式具有独立性，不受其他步骤影响；
每步所取的方式不同，不会得出（整体的）相同方式。

[EXAMPLE] Diane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible outfits.

[ANSWER] $2\times 4\times 2=16$