Probability · 概率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| independent/ˌɪndɪˈpendənt/ | 独立 | dú lì |
| mutually exclusive/ˈmjuːtʃuːəli eksˈkluːsɪv/ | 互斥 | hù chì |
| conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ | 条件概率 | tiáo jiàn gài lǜ |
| tree diagram/triː ˈdaɪəɡræm/ | 树状图 | shù zhuàng tú |
| without replacement/wɪˈðaʊt rɪˈpleɪsmənt/ | 不放回 | bù fàng huí |
| Venn diagram/ven ˈdaɪəɡræm/ | 韦恩图 | wéi ēn tú |
| two-way table/tuː weɪ ˈteɪbl/ | 双向表 | shuāng xiàng biǎo |
The birthday paradox
- In a room of just 23 people, there's a greater than 50% chance that two share a birthday.
- Probability is counterintuitive. The rules are simple, but the results can surprise you.
生日悖论
- 在一个只有 23 人的房间里,有超过 50% 的机会两个人共享一个生日。
- 概率是反直觉的。规则简单,但结果可能让你吃惊。
Combining probabilities
- Addition ("or"): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
- Multiplication ("and", if independent 独立): $P(A \cap B) = P(A)\,P(B)$.
- Mutually exclusive 互斥 events can't both happen; independent events don't affect each other.
Worked example. P(rain) = 0.3, P(bus late) = 0.2, independent. P(rain or bus late) = 0.3 + 0.2 − 0.3 × 0.2 = 0.44.
Mutually exclusive ≠ independent. Mutually exclusive means they can't both happen (so $P(A \cap B) = 0$). Independent means one doesn't affect the other (so $P(A \cap B) = P(A) \times P(B)$). These are completely different concepts.
Dice: a starting point for probability
组合概率
- 加法(“或”):$P(A \cup B) = P(A) + P(B) - P(A \cap B)$。
- 乘法(“且”,如果独立):$P(A \cap B) = P(A)\,P(B)$。
- 互斥(mutually exclusive)事件不能都发生;独立(independent)事件互不影响。
算例。 P(下雨) = 0.3,P(公交晚点) = 0.2,独立。P(下雨或公交晚点) = 0.3 + 0.2 − 0.3 × 0.2 = 0.44。
互斥 ≠ 独立。 互斥意味着它们不能都发生(所以 $P(A \cap B) = 0$)。独立意味着一个不影响另一个(所以 $P(A \cap B) = P(A) \times P(B)$)。这是完全不同的概念。
骰子:概率的一个起点
Conditional probability · 条件概率
P(A ∩ B) = P(A)·P(B|A)
The area model shows how conditional probabilities combine — and how Bayes finds P(A | B). · 面积模型显示条件概率如何组合——以及贝叶斯如何求 P(A | B)。
P(A) = 0.5, P(B) = 0.4 and P(A∩B) = 0.2. Are A and B independent? · P(A) = 0.5,P(B) = 0.4,P(A∩B) = 0.2。A 和 B 独立吗?
P(A)×P(B) = 0.5 × 0.4 = 0.2, which equals P(A∩B), so they are independent. · P(A)×P(B) = 0.5 × 0.4 = 0.2,它等于 P(A∩B),所以它们独立。
P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.2. What is P(A∪B)? · P(A) = 0.5,P(B) = 0.4,P(A∩B) = 0.2。P(A∪B) 是多少?
P(A∪B) = 0.5 + 0.4 − 0.2 = 0.7. · P(A∪B) = 0.5 + 0.4 − 0.2 = 0.7。
If A and B are mutually exclusive, then P(A∩B) = 0. · 如果 A 和 B 互斥,那么 P(A∩B) = 0。
Mutually exclusive events cannot both happen, so their intersection has probability 0. · 互斥的事件不能都发生,所以它们的交集概率为 0。
Mutually exclusive events are always independent. · 互斥的事件总是独立的。
If A and B are mutually exclusive (and both have P > 0), then P(A∩B) = 0 ≠ P(A)×P(B), so they are NOT independent. · 如果 A 和 B 互斥(且两者 P > 0),那么 P(A∩B) = 0 ≠ P(A)×P(B),所以它们不独立。
Conditional probability 条件概率
- Conditional: the chance of $A$ given $B$ has happened:
- Independence test: check whether $P(A \cap B) = P(A)\times P(B)$.
The normal curve: a probability is the area under it, centred on the mean
条件概率
- 条件(conditional):在 $B$ 已发生的情况下 $A$ 的机会:
- 独立性检验:检查是否 $P(A \cap B) = P(A)\times P(B)$。
正态曲线:一个概率是它下面的面积,以平均数为中心
P(A∩B) = 0.2 and P(B) = 0.4. What is P(A | B)? · P(A∩B) = 0.2 而 P(B) = 0.4。P(A | B) 是多少?
P(A|B) = P(A∩B)/P(B) = 0.2/0.4 = 0.5. · P(A|B) = P(A∩B)/P(B) = 0.2/0.4 = 0.5。
Worked example — tree diagrams 树状图
- A bag has 3 red and 2 blue balls. Two are drawn without replacement 不放回.
- P(both red) = $\dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{6}{20} = 0.3$.
- P(one of each) = $\dfrac{3}{5} \times \dfrac{2}{4} + \dfrac{2}{5} \times \dfrac{3}{4} = \dfrac{6}{20} + \dfrac{6}{20} = 0.6$.
On a tree diagram, multiply probabilities along the branches, then add the paths you want.
算例——树形图
- 一个袋里有 3 个红球和 2 个蓝球。不放回地抽两个。
- P(都是红) = $\dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{6}{20} = 0.3$。
- P(各一个) = $\dfrac{3}{5} \times \dfrac{2}{4} + \dfrac{2}{5} \times \dfrac{3}{4} = \dfrac{6}{20} + \dfrac{6}{20} = 0.6$。
在一张树形图上,沿分支把概率相乘,然后把你要的路径相加。
A bag has 3 red and 2 blue. Two drawn without replacement. P(both red) = (3/5)(2/4). Find it. · 一个袋里有 3 个红的和 2 个蓝的。不放回地抽两个。P(都是红) = (3/5)(2/4)。求它。
(3/5) × (2/4) = 6/20 = 0.3. · (3/5) × (2/4) = 6/20 = 0.3。
A bag has 3 red and 2 blue, drawn without replacement. Put the steps for P(both red) in order. · 一个袋里有 3 个红的和 2 个蓝的,不放回地抽。把求 P(都是红) 的步骤按顺序排列。
On a tree, multiply the probabilities along the chosen branch, then simplify. · 在一棵树上,沿所选的分支把概率相乘,然后化简。
Venn diagrams 韦恩图 and two-way tables 双向表
- Venn diagrams show overlapping events visually — the intersection is $A \cap B$.
- Two-way tables organise data by two categories — read conditional probabilities from the relevant row or column.
- Count arrangements with permutations (order matters) and combinations (order does not).
维恩图与双向表
- 维恩图(Venn diagrams)直观地显示重叠的事件——交集是 $A \cap B$。
- 双向表(two-way tables)按两个类别组织数据——从相关的行或列读条件概率。
- 用排列(permutations,有序)和组合(combinations,无序)计数。
You've got it
- addition (or): $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
- multiplication (and, independent): $P(A\cap B) = P(A)P(B)$
- conditional: $P(A\mid B) = \dfrac{P(A\cap B)}{P(B)}$; test independence with $P(A\cap B) = P(A)P(B)$
你掌握了
- 加法(或):$P(A\cup B) = P(A) + P(B) - P(A\cap B)$
- 乘法(且,独立):$P(A\cap B) = P(A)P(B)$
- 条件:$P(A\mid B) = \dfrac{P(A\cap B)}{P(B)}$;用 $P(A\cap B) = P(A)P(B)$ 检验独立性