Probability
| English | Chinese | Pinyin |
|---|---|---|
| probability | 概率 | gài lǜ |
| mutually exclusive | 互斥 | hù chì |
| independent | 独立 | dú lì |
| complement | 对立事件 | duì lì shì jiàn |
| permutation | 排列 | pái liè |
| combination | 组合 | zǔ hé |
How likely, on a scale of nothing to certain
- Probability 概率 puts a number between 0 and 1 on how likely something is.
- For equally likely outcomes it is favourable outcomes over total outcomes — which makes counting the real work.
- Most probability errors in this module are counting errors, not probability errors.
Combining events
- Events are mutually exclusive 互斥 when they cannot both happen: $P(A \text{ or } B) = P(A) + P(B)$.
- Events are independent 独立 when one does not affect the other: $P(A \text{ and } B) = P(A) \times P(B)$.
- ⚠ "Or" adds, "and" multiplies — but only under those conditions. Drawing without replacement breaks independence.
Independent events, combined
See why "and" multiplies while "or" adds — and when each rule applies.
The complement
- The complement 对立事件 rule: $P(\text{not } A) = 1 - P(A)$.
- It is usually the fastest route to anything phrased "at least one".
- Counting the ways to get at least one red ball is long. Counting the one way to get no red ball is short.
A question asks for P(at least one). What is usually the fastest route?
There is usually one way to get none and many ways to get at least one.
A bag holds 4 red and 6 blue balls. Two are drawn without replacement. Find P(at least one red).
Both the numerator and the denominator fell on the second draw, because a ball was removed. Forgetting to change the denominator is the standard error, and it is the whole difference between "with" and "without replacement".
4 red and 6 blue balls, two drawn without replacement. What is P(at least one red)?
P(no red) = 6/10 × 5/9 = 1/3, so the complement is 2/3.
Drawing without replacement leaves the two draws independent.
The first draw changes what is left, so the second probability depends on it.
Counting techniques
- A permutation 排列 counts ordered selections; a combination 组合 counts unordered ones.
- A committee is a combination. A podium of first, second and third is a permutation.
- Ask "does the order change the outcome?" — that one question chooses the formula.
Choosing 3 students for a committee from 10. Permutation or combination?
The order of a committee does not change who is on it, so order does not matter.
Read for "with replacement" or "without". It changes every fraction after the first one. Underlining those three words before you calculate is worth more marks in this unit than any formula.
Probabilities cannot exceed 1. If your answer does, you almost certainly added when you should have multiplied, or added two events that can happen together. Checking that the total is between 0 and 1 costs nothing and catches both mistakes.
A calculated probability of 1.2 means the event is very likely.
It means the calculation is wrong. Probabilities never exceed 1 — usually you added instead of multiplying.