Introduction to Probability · 概率入门
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| sample space/ˈsæmpl speɪs/ | 样本空间 | yàng běn kōng jiān |
| outcome/ˈaʊtkʌm/ | 结果 | jié guǒ |
| complement rule/ˈkɒmplɪmənt ruːl/ | 补集法则 | bǔ jí fǎ zé |
| equally likely/ˈiːkwəli ˈlaɪkli/ | 等可能 | děng kě néng |
| Venn diagram/ven ˈdaɪəɡræm/ | 韦恩图 | wéi ēn tú |
All the ways it could go
- The sample space 样本空间 is the set of all possible outcomes of a chance process.
- Each individual result is an outcome 结果; an event is a set of outcomes.
- Rolling a die: sample space $\{1,2,3,4,5,6\}$; the event "even" $= \{2,4,6\}$.
- Listing the sample space is the first step in finding a probability.
所有可能的走向
- 样本空间是一个机会过程所有可能结果的集合。
- 每个单独的结果是一个结果；一个事件是一组结果。
- 掷一个骰子：样本空间 $\{1,2,3,4,5,6\}$；事件“偶数” $= \{2,4,6\}$。
- 列出样本空间是求概率的第一步。
The basic rules
- Every probability satisfies $0 \le P(A) \le 1$ (never negative, never above $1$).
- The probabilities of all outcomes in the sample space sum to $1$.
- Complement rule 补集法则: $P(A^c) = 1 - P(A)$ — "not $A$" is whatever's left.
- These rules constrain every valid probability model.
基本法则
- 每个概率都满足 $0 \le P(A) \le 1$（永不为负，永不超过 $1$）。
- 样本空间中所有结果的概率之和为 $1$。
- 补集法则：$P(A^c) = 1 - P(A)$——“非 $A$”就是剩下的部分。
- 这些法则约束着每一个有效的概率模型。
Equally likely outcomes
- When outcomes are equally likely 等可能, probability is just counting.
- $P(A) = \dfrac{\text{number of outcomes in } A}{\text{total number of outcomes}}$.
- $P(\text{even on a die}) = 3/6 = 0.5$.
- This "favorable over total" rule works only when outcomes are equally likely.
等可能结果
- 当结果是等可能的，概率就只是计数。
- $P(A) = \dfrac{\text{favorable}}{\text{total}}$（$A$ 中的有利结果数 ÷ 结果总数）。
- $P(\text{even on a die}) = 3/6 = 0.5$。
- 这个“有利数除以总数”的法则只在结果等可能时才成立。
Picturing events
- A Venn diagram 韦恩图 draws events as overlapping circles inside the sample space.
- The overlap is the intersection (both events); the whole shaded region is the union (either).
- A two-way table does the same job with counts in rows and columns.
- Both make "and," "or," and "not" easy to see and compute.
画出事件
- 维恩图把事件画成样本空间内相互重叠的圆。
- 重叠部分是交集（两个事件同时）；整个阴影区域是并集（其中之一）。
- 双向表用行和列里的计数做同样的事。
- 两者都让“且”“或”“非”易于看清和计算。
The counting rule $P(A)=\frac{\text{favorable}}{\text{total}}$ only works when outcomes are equally likely. For a biased coin or unequal outcomes, you must use the given probabilities, not just count. And always check your model obeys the basics: every $P$ in $[0,1]$, and the total over the sample space equal to $1$.
计数法则 $P(A)=\frac{\text{favorable}}{\text{total}}$ **只在结果等可能时才成立。**对于有偏的硬币或不等可能的结果，你必须使用给定的概率，而不能只是计数。并且总要检查你的模型遵守基本法则：每个 $P$ 都在 $[0,1]$ 内，且在样本空间上的总和等于 $1$。
Draw one card from a standard $52$-card deck.
- Sample space: the $52$ cards (equally likely). Event "heart" has $13$ outcomes.
- $P(\text{heart}) = 13/52 = 0.25$.
- Complement: $P(\text{not a heart}) = 1 - 0.25 = 0.75$.
从一副标准的 $52$ 张牌中抽一张。
- **样本空间：**这 $52$ 张牌（等可能）。事件“红心”有 $13$ 个结果。
- $P(\text{heart}) = 13/52 = 0.25$。
- 补集：$P(\text{not a heart}) = 1 - 0.25 = 0.75$。
The sample space is all possible outcomes; every model has $0 \le P(A) \le 1$ with the total equal to $1$, and the complement rule $P(A^c) = 1 - P(A)$. For equally likely outcomes, $P(A) = \frac{\text{favorable}}{\text{total}}$. Represent events with a Venn diagram or two-way table.
样本空间是所有可能的结果；每个模型都有 $0 \le P(A) \le 1$ 且总和为 $1$，以及补集法则 $P(A^c) = 1 - P(A)$。对等可能结果，$P(A) = \frac{\text{favorable}}{\text{total}}$。用维恩图或双向表表示事件。
Events as overlapping circles · 用重叠的圆表示事件
A Venn diagram shows unions, intersections, and complements. · 维恩图展示并集、交集和补集。
Rolling a fair die, what is P(even)? Give a decimal. · 掷一个均匀骰子，P(偶数) 是多少？用小数。
Even = {2,4,6}, so 3/6 = 0.5. · 偶数 = {2,4,6}，所以 3/6 = 0.5。
If P(A) = 0.25, what is P(not A) by the complement rule? · 若 P(A) = 0.25，由补集法则，P(非 A) 是多少？
P(A^c) = 1 − 0.25 = 0.75. · P(A^c) = 1 − 0.25 = 0.75。
The rule P(A) = favorable/total works only when the outcomes are... · 法则 P(A) = 有利数/总数 只在结果……时才成立。
Counting works only for equally likely outcomes. · 计数只对等可能结果有效。
A valid probability can be 1.4. · 一个有效的概率可以是 1.4。
Every probability must satisfy 0 ≤ P ≤ 1. · 每个概率都必须满足 0 ≤ P ≤ 1。
The set of all possible outcomes of a chance process is the sample ___. · 一个机会过程所有可能结果的集合是样本 ___（填英文一词）。
The sample space lists every possible outcome. · 样本空间列出每一个可能的结果。