Probability
Basic probability: certain, impossible and random events, sample spaces, and listing outcomes with tables and tree diagrams, with expected value as an NF extension. It is the base of F5 more about probability.
What you will learn
- 31.1 Certain, impossible and random events.
- 31.2 Probability; sample space. Venn diagrams may be used. Geometric probability is not required.
- 31.3 Find probabilities by listing the sample space with tables or tree diagrams.
- 31.4 Problems involving probability.
- NF 31.5–31.6 Expectation, and problems using it.
You need first: Number and algebra basics
Key terms
| English | 中文 |
|---|---|
| event | 事件 |
| certain / impossible event | 必然事件／不可能事件 |
| random | 隨機 |
| probability | 概率 |
| sample space, outcome | 樣本空間，結果 |
| equally likely | 機會均等 |
| tree diagram | 樹形圖 |
| with / without replacement | 放回／不放回 |
| expected value | 期望值 |
Standard steps
Basic probability
P(E) = number of favourable outcomes ÷ total number of equally likely outcomes.
- 0 ≤ P(E) ≤ 1.
- P(not E) = 1 − P(E).
Two-step experiments
- Table when both steps have the same outcomes (two dice): fill a 6 × 6 grid and count.
- Tree diagram otherwise:
- One set of branches per step. Write the probability on each branch.
- Multiply along a path.
- Add the paths that give the event.
- Without replacement: on the second step, the total goes down by 1, and so does the count of whatever was taken.
Expectation (NF)
E = Σ (value × probability). For a game, net gain = expected winnings − the cost to play.
Worked examples
Favourable outcomes: (2,6), (3,5), (4,4), (5,3), (6,2), so 5 of 36. P = 5/36.
P(RR) = 3/5 × 2/4 = 6/20
P(BB) = 2/5 × 1/4 = 2/20
P(same colour) = 6/20 + 2/20 = 8/20 = 2/5expected winnings = 50 × 1/8 + 20 × 1/4 = 6.25 + 5 = $11.25
expected net gain = 11.25 − 10 = $1.25 per gameCommon mistakes
- Treating outcomes that are not equally likely as if they were ("sum 2 and sum 7 are equally likely").
- Not reducing the total on the second draw without replacement.
- Adding along a branch instead of multiplying.
- Forgetting the reverse order: "one red and one blue" is RB or BR.
- Expectation without subtracting the cost of playing.
How the DSE tests it
- Paper 2 Section A: dice, cards, bags of balls, tables of results.
- Paper 1: probability inside a statistics question ("a student is chosen at random from the table…").
- F5 builds on this with combinations and conditional probability.
Questions you might not ask in class
Why multiply along branches?
If 3 in 5 times you get red first, and then 2 in 4 of those times you get red again, that is 2/4 of 3/5, and "of" means multiply.
Expected value $11.25, but you can't win $11.25 in one game.
Right. It is the average per game if you played many, many times.
Is 'random' the same as 'equally likely'?
Not always. A random choice from a fair bag is equally likely; a random day's weather is not.