Random and Non-Random Patterns · 随机与非随机模式
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| randomness/ˈrændəmnəs/ | 随机性 | suí jī xìng |
| probability/ˌprɒbəˈbɪlɪti/ | 概率 | gài lǜ |
| chance/tʃæns/ | 偶然 | ǒu rán |
| long-run relative frequency/lɒŋ rʌn ˈrelətɪv ˈfriːkwənsi/ | 长期相对频率 | cháng qī xiāng duì pín lǜ |
Order out of chaos
- A single random outcome is unpredictable — but many of them form a stable pattern.
- Randomness 随机性 means individual results vary and can't be foreseen one at a time.
- Yet over many repetitions, the proportions settle down to something predictable.
- Probability is the math of that long-run stability.
混沌中的秩序
- 单个随机结果无法预测——但许多个结果会形成一个稳定的模式。
- 随机性意味着个别结果各不相同，无法逐个预见。
- 然而在多次重复之后，各种结果的比例会稳定到可预测的水平。
- 概率就是研究这种长期稳定性的数学。
Could it be just chance?
- The core question: is a result a real effect, or could it happen by chance 偶然?
- Random variation alone can produce streaks and clusters that look meaningful.
- To judge "surprising," we compare what we saw against what chance would do.
- If chance can easily produce it, it isn't strong evidence of anything.
会不会只是碰巧？
- 核心问题：一个结果是真实效应，还是可能出于机会？
- 仅凭随机变异，就能产生看起来有意义的连胜和聚集。
- 要判断是否“意外”，我们把看到的与机会会产生的作比较。
- 如果机会能轻易产生它，它就不是任何有力的证据。
Probability = long-run frequency
- Probability 概率 is the long-run relative frequency 长期相对频率 of an outcome.
- $P(\text{heads}) = 0.5$ means: over many tosses, about half land heads.
- It's not a promise about the next toss — it's about the long run.
- A probability is always a number between $0$ and $1$.
概率 = 长期频率
- 概率是一个结果的长期相对频率。
- $P(\text{heads}) = 0.5$ 意味着：在许多次投掷中，大约一半正面朝上。
- 它不是对下一次投掷的承诺——它讲的是长期。
- 概率永远是 $0$ 到 $1$ 之间的一个数。
Why we need a model
- To decide if data are surprising, we need a probability model to compare against.
- The model says what outcomes chance should produce, and how often.
- Then "surprising" = far from what the model predicts.
- Units 4–5 build these models so later units can test claims.
为什么需要一个模型
- 要判断数据是否意外，我们需要一个概率模型来作比较。
- 模型说明机会应该产生哪些结果、以及多频繁。
- 于是“意外” = 远离模型的预测。
- 第 4–5 单元建立这些模型，好让后面的单元能检验主张。
Probability describes the long run, not the next trial. "$P=0.5$" does not mean heads and tails must alternate, or that a run of $5$ heads is "due" to end — each toss is fresh. The law of averages ("I'm due for a win") is a fallacy; chance has no memory. The stability appears only over many repetitions.
概率描述的是长期，而非下一次试验。“$P=0.5$”不意味着正反面必须交替出现，也不意味着连出 $5$ 个正面后“该”结束了——每次投掷都是全新的。平均律（“我该赢了”）是一种谬误；机会没有记忆。稳定性只在许多次重复之后才显现。
Flip a fair coin.
- Short run: $5$ flips could easily be HHHHH — chance is streaky.
- Long run: over $10{,}000$ flips, the fraction of heads is very close to $0.5$.
- So $P(\text{heads}) = 0.5$ is a statement about the long run, not any single flip.
抛一枚均匀硬币。
- 短期：$5$ 次抛掷很容易出现 HHHHH——机会是有连贯性的。
- **长期：**在 $10{,}000$ 次抛掷中，正面的比例非常接近 $0.5$。
- 所以 $P(\text{heads}) = 0.5$ 是关于长期的陈述，而非任何单次抛掷。
Randomness makes individual outcomes vary unpredictably, but over many repetitions the relative frequency stabilizes. Probability is that long-run relative frequency, always between $0$ and $1$. We need a probability model to judge whether an observed result is surprising or just chance.
随机性让个别结果不可预测地变动，但在多次重复后相对频率会稳定下来。概率就是那个长期相对频率，永远在 $0$ 到 $1$ 之间。我们需要一个概率模型来判断观察到的结果是意外的还是仅仅出于机会。
A probability from 0 to 1 · 0 到 1 之间的一个概率
Probability lives on a scale from 0 (impossible) to 1 (certain). · 概率位于从 0（不可能）到 1（必然）的刻度上。
Probability is best described as the long-run... · 概率最好被描述为长期的……
Probability = long-run relative frequency over many repetitions. · 概率 = 多次重复中的长期相对频率。
After 5 heads in a row on a fair coin, tails is 'due' and more likely on the next flip. · 均匀硬币连出 5 个正面后，反面“该来了”，下一次更可能是反面。
Each flip is independent — chance has no memory. That's the law-of-averages fallacy. · 每次抛掷都是独立的——机会没有记忆。这是平均律谬误。
What is the largest value a probability can take? · 概率能取到的最大值是多少？
Probabilities range from 0 to 1. · 概率的范围是 0 到 1。
The property that individual outcomes vary and can't be predicted one at a time is called ___. · 个别结果各不相同、无法逐个预测的性质叫做 ___（填英文一词）。
Randomness produces short-run variability but long-run stability. · 随机性产生短期变异但长期稳定。
Why do we need a probability model? · 我们为什么需要一个概率模型？
The model says what chance should do, so we can spot the surprising. · 模型说明机会应该做什么，于是我们能识别意外。